On Collatz Problem

[Organic]

Collatz problem can be found here: http://mathworld.wolfram.com/CollatzProblem.html



Please look at the attached paper:

http://www.geocities.com/complementarytheory/3n1proof.pdf


and reply your comments.


Thank you,


Organic

[matt grime]

Comments1. k must be fixed one presumes or many things make no sense as we will see.

2. The second line, starting 'A direct convergence', has an extraneous comma that makes it unclear what you mean


3. the 4th paragraph implies that k is not fixed, since if it were 2^{k+1} would contradict that statement.

4.of the 4 options you give one must conclude that k is fixed otherwise the assertion j is an even number >2^k is nonsense. It is bad maths anyway, as one should not use symbols when words are required.

5. similarly don't use XOR like that, neither of the 'inputs' is a statement than can be true or false.

are those contradictions about k being fixed or not enough for you? probably not.


edit:

there are also these:


what does it mean 'to be out of the range'?


why are you misusing decidability like this? godel states that something is undecidable if both it and its negation are consistent with the other axioms.


i don't even want to touch the von neumann heirarchy stuff.

[matt grime]

thinking about this for a while it just seems that you're saying: because of a couple of iterations of the rules of Collatz we see it can never be decided what happens. but that is obviously wrong.

[Organic]

Hi Matt,

Thank you for your reply.


k must be fixed one presumes or many things make no sense as we will see.

k is any positive integer ( 0 included).

2. The second line, starting 'A direct convergence', has an extraneous comma that makes it unclear what you mean

Thank you, you right (my English problems again), A direct convergence to 1 must start in one of the even numbers that belong to 2^k sequence.

3. the 4th paragraph implies that k is not fixed, since if it were 2^{k+1} would contradict that statement.

As i wrote, k is any positive integer ( 0 included).

4.of the 4 options you give one must conclude that k is fixed otherwise the assertion j is an even number >2^k is nonsense. It is bad maths anyway, as one should not use symbols when words are required.

Please explain what do you mean by "k is fixed"?, again k is any positive integer ( 0 included).

5. similarly don't use XOR like that, neither of the 'inputs' is a statement than can be true or false.

If i write "or" instead of "XOR" is it ok?

what does it mean 'to be out of the range'?

If you look at the Binary Tree that stands in the base of Von Neumann Hierarchy, then you can see that this structure is the invariant symmetry of N members, therefore can be used to define rigorous proofs about N members.

Collatz problem is also based on this invariant symmetry of the Binary Tree, and for any k we can find N members that are out of the range of 2^k (case 4 in page 1).

Therefore it is non-decidable for any Binary Tree, which means non-decidable within N (for any n).

[matt grime]

I don't think you understand the subtlety of the word fixed.


As you compare j to 2^k it cannot be that when you say 'examine 2^k' that you mean 'examine the set {2^k | k in N u{0}}'

you are confusing sets and numbers again and the comparison j>2^k is meaningless without explanation.

I really don't see what you are getting at, to be honest, at least beyond stating in a very confused fashion the obvious about what might happen when you apply the Collatz iteration.


Anyway, that can be readily rectified, however you don't explain what you mean by 'out of the range' no matter what you may think. A simple statement will do, you know. Nor do you explain how your facile observations imply anything to do with decidability, which as I pointed out is quite an intricate issue - the continuum hypothesis is undecidable in ZFC for instance.


or is better than xor, but you shouldnt write 'either >x or <x' in a sentence as that is deemed poor presentation.

[Organic]

Dear Matt,

Thank you very much for your help.

I fixed some of the problems.

Please read it again:

http://www.geocities.com/complementarytheory/3n1proof.pdf

and please tell me more about "fixed" if needed.

Please be aware that i am using here a non-standard point of view on N members, which based on its fundamental symmetry of the Binary Tree.

Yours,

Organic

[matt grime]

changing the word range to the word scope doesn't make it any clearer. just state what you mean to be out of the scope of, or out of the range of

[Organic]

Well, I used scope because "range" is already used by Math.

By out of the scope I mean that we shall always find n's which are beyond any Binary Tree.

Because this symmetry stands in the basis of N members, we can conclude that Collatz Problem is non-decidable for any n in N, because any n is nothing but a part of the Binary Tree invariant symmetry.

I'm going to sleep, see you tomorrow.

[matt grime]

so you assertion is that given any binary tree there is a number 'beyond' it. Care to explain what that means at any point?

anyway, you've not proved that at no point in the future iteration of 47 does it not return to one. I'll put money on it doing so.

Putting it in your language, you've omitted to consider things in the correct order, which is that given an n there is some k with n in the scope of the binary tree 2^k, and in fact in the scope of infinitely many of them. You're taking things in the wrong order again.

Of course if you could prove that there is some n, such that for any k, there is some point in the Collatz iteration with i_r(n)>2^k, where i_r stands for the r'th iterate in the sequence, but that isn't what you've shown.


The assertion that given any k there is an even integer, n, not a power of 2, with n>2^k does not prove anything interesting here.


Let's repeat the main point. There is no number beyond the scope of any binary tree.

Given any binary tree there is a number beyond its scope, but that is different for each tree. There is no number that is beyond the scope of EVERY tree.

Not that I see what that would prove anyway.

[Organic]

Hi Matt,

I made major update in my proof, please tell me what do you think:

http://www.geocities.com/complementarytheory/3n1proof.pdf

Thank you,

Organic

[matt grime]

The assertion that, given a k, 3*(2^k+1)+1 is not in *any* binary tree is wrong. It is in an infinite number of them. It is in the one corresponding to 2^r where r is any number greater than 3*(2^k+1)+1.

When you say the first number after 2^k you are fixing k at some (arbitrary) value.

because, once you've fixed this k, you can find a number iterated to some larger number is not important because it is dependent on k, only if it were independent would you even start to have something, but even then that is moot as you'd have to prove that no iteration at any point in the future would return you to 1

example k=3, so we start with 9, the iterations are

9,28,14,7,22,11,34,17,52,26,13,40,20,10,5,16,8,4,2 ,1



It still wouldn't state it was undecidable:

a statement is undecidable in some axiomatic theory if either the assumption of it or its negation is consistent with the axioms, example, the continuum hypothesis and ZFC.

[Organic]

Hi Matt,

But what I show is based on the invariant symmetry that stands in the base of any n in N.

(2^k)+1 has a 1-1 and onto with {} that exists in the internal structure (The Von Neumann Hierarchy Binary Tree) of any n in N.

Let current n be 2^(k+1).

Any 3*((2^k)+1)+1(=3n+1) > 2^(k+1) where 3n+1 is beyond the internal Binary Tree structure of current n(=2^(k+1)) in N, which means that we always need to get n+1 for general and rigorous proof. Shortly speaking, if n then n+1 (the ZF axiom of infinity), therefore it is non-decidable for any n in N.
Q.E.D

Please see my updates:

http://www.geocities.com/complementarytheory/3n1proof.pdf

[Organic]

This self similarity of N members, which constructed by The Von Neumann Hierarchy Binary Tree can't be ignored.

Maybe someone knows what is the Mathematical brunch that researches the symmetry of Mathematical objects like numbers?

[matt grime]

Originally posted by Organic
Hi Matt,

But what I show is based on the invariant symmetry that stands in the base of any n in N.

(2^k)+1 has a 1-1 and onto with {} that exists in the internal structure (The Von Neumann Hierarchy Binary Tree) of any n in N.

Let current n be 2^(k+1).

Any 3*((2^k)+1)+1(=3n+1) > 2^(k+1) where 3n+1 is beyond the internal Binary Tree structure of current n(=2^(k+1)) in N, which means that we always need to get n+1 for general and rigorous proof. Shortly speaking, if n then n+1 (the ZF axiom of infinity), therefore it is non-decidable for any n in N.
Q.E.D

Please see my updates:

http://www.geocities.com/complementarytheory/3n1proof.pdf



Look, the number you pick for each k that is out of the 'scope' of the tree you are looking at is different for each tree, there is not universal number in those that you pick. Got it? Evidently not. You fix the k, then you say take 3*(2^k+1)+1, note that is different for each k and there is not one of them you can pick that works for all k.


For pity's sake don't start using the axoim of infinity again, as this isn't what it states. It merely states the the set of integers must be a set in the model you are using.

You are evidently using your own private defintion of undecidable which is not the mathematical one (again).


In particular all it appears you are saying is that there is an odd number freater than 2^k. That number you chose is different for each k. The axiom of infinity that you seem obsessed with doesn't all ow you take the limit of this sequence and claim it is a meaningful number. This is just the same as the fundamental error in your cantor argument.

What you've written down does not support any conclusions you've made.

[Organic]

Matt,

Again you ignore the invariant fundamental symmetry that stands in the basis of N.

My proof is not about any fixed k value. It is about the "universal" (your word) structure that stands in the basis of N.

What I show is that there is self similarity between this basic structure and the Binary Tree used in Collatz problem.

Therefore if there is a general and rigorous proof to Collatz problem, it means that you also prove the ZF axiom of infinity ("if n exists then n+1 exists" is an axiom, therefore cannot be proved, otherwise it is not an axiom).

Shortly speaking, a general and rigorous proof to Collatz problem is a contradiction of ZF axiom of infinity existence as an axiom.

Because no proof of any Math system contradicts its axioms, Collatz problem is undecidable within N.

Also sets R and C can't be used here because their existence depends on set N existence.

My proof is "structured oriented", not "fixed k oriented".


About |N| and Collatz problem

If we look at this tree http://michael.cleverly.com/funstuff/3x+1/collatz2.jpg we can see that there are some numbers which are the results of both 3n+1 and n/2, for example:

Number 22 is the result of 3*7+1 and also the result of 44/2.

Let K be the set off all natural numbers, which are a common result to both 3n+1 and n/2.

|K|=|N| therefore transfinite system is too strong for Collatz problem.


Arithmetic

The arithmetic of Collatz problem is not importent, because we can use another arithmetic and get the same "Collatz tree" structure with different values instead of N members.

Here is your example:

Define the 'arithmetic' in anyway you choose that happens to agree with that specific example on those particular numbers. Eg, the Collatz rules except when a_n=1,000,000,000, and anything but division by two there.

If you mean is there another way of defining a Collatz type relation which agrees on ALL values of a_n, then the answer is yes and no. There are equivalent constructions, but functionally they are the same. By that I mean

sin(x) and cos(x-pi/2) are functionally the same, but have different 'labels'

[matt grime]

Oh, not again.


I ignore the fundamental symmetry you claim is there because you've not demonstrated it proves anything.

You have fixed k, because you say consider 3*(2^k+1)+1. That doesn't meant to say you've set k to be equal to some number. You then say that becase this isn't in the 'scope' of the binary tree for 2^k (still without defining scope) which I take to mean is bigger than something. However for each k you get a different thing. I've already shown you 2^k+1 goes to 1 for k=3, it does for 2 as well. It shouldn't be beyond you to check that it does for 4, 5 and 6 as well. So what's the point you're trying to make?


The set N exists. It is in a model for ZF because of the axiom of infinity. If you wish to do a set theory omiting the axiom of infinity then N will not necessarily be a set in that theory.


How have you concluded the Collatz problem contradicts ZF? How?


The word universal was only used in an attempt to describe what it was you were claiming about these 2^k+1 things.

You've misused the axiom of infinity again. DOes this mean you've finally accepted that the sentence

If n then n+1

makes no sense either mathematically or in English?

If X then Y is a proposition, ther terms X and Y must be statements that are either true or false.

I don't think there can be a 'self'-similarity between two different things, btw. And what boolean tree for Collatz? (I'm not sure I've ever heard the term boolean tree before) and even if there were these things you've not demonstrated anything. You'd think the fact that the numbers you're using to contradict the axiom of infinity using COollatz wouldn't so clearly reach 1 so quickyl for small k, but they do, don't they?

So what that there the cardinality of the set of numbers that are obtained from more than one possible starting point are the same: Consider the itereation defined by n-->1 for all n

every number ends up at 1 after one interation. By your logic it is undecidable in ZF if every number ends up at 1. After all, the set of elements which trivally get to 1 in COllatz is also uncountable too, the powers of 2 for instance.

The quote you posted from me is irrelevant as it was answering an unrealted question you posed: are there other ways of defining the opertations that agree on a certain finite subset of the naturals with the COllatz seqence? The arithmetic of the Collatz sequence IS important as it must agree for all integers. I was also indicating there might be a way to extend it to a function of the complex plane that agreed with COllatz on the integers - like the gamma and factorial functions.

[Organic]

Matt,

A simple qeustion:

Complex numbers existence, does it depends on N numbers existence?

You use the words "agree with".

Well this is exactly what I mean by saying that the important thing here is the invariant "Collatz Tree" and not its 'labels' that can be N, Q, R or C 'labels'.


You CAN'T ignore the invariant fundamental symmetry of the Binary Tree that stands in the basis of set N members.

What I show is that there is self similarity between this basic structure and the Binary Tree used in Collatz problem.

Therefore if there is a general and rigorous proof to Collatz problem, it means that you also prove the ZF axiom of infinity ("if n exists then n+1 exists" is an axiom, therefore cannot be proved, otherwise it is not an axiom).

Shortly speaking, a general and rigorous proof to Collatz problem is a contradiction of ZF axiom of infinity existence as an axiom.

Because no proof of any Math system contradicts its axioms, Collatz problem is undecidable within N.

Also sets R and C can't be used here because their existence depends on set N existence.

My proof is "structured oriented" and if you don't get it you can't understand it.

[matt grime]

Originally posted by Organic
Matt,


You CAN'T ignore the invariant fundamental symmetry of the Binary Tree that stands in the basis of set N members.

You would have to demonstrate that this alleged fundamenetal symmetry has any bearing on the case in hand

Therefore if there is a general and rigorous proof to Collatz problem, it means that you also prove the ZF axiom of infinity ("if n exists then n+1 exists" is an axiom, therefore cannot be proved, otherwise it is not an axiom).

I don't think you understand what you're talking about here. The collection of elements of N exists, ZFC has an axiom that means that it constitutes a set, that's all. Or another way, it tells us there must be an inductive set in our model. The Naturals form this set in our model.

It is very unclear what you are attempting to claim the contradiction is here. If one proves the Collatz conjecture one proves the naturals exist? They exist anyway, and happen to constiute a set in ZFC. So what?

Shortly speaking, a general and rigorous proof to Collatz problem is a contradiction of ZF axiom of infinity existence as an axiom.

no it isn't

Because no proof of any Math system contradicts its axioms, Collatz problem is undecidable within N.

Also sets R and C can't be used here because their existence depends on set N existence.

I won't and don't use R or C here.

Anyway, as you claim to have demonstrated the Collatz conjecture in itself contradicts the axioms of ZF, it is not that it is undecidable, but that it is false. Or at least it would be that if you assumed it true, then it contradicts ZFC hence it is false. So well, done you've disproved the Collatz conjecture, allegedly. Of course you haven't cos your proof is utterly wrong.

Undecidable means that both it and its negation must be consistent with the axioms.

[Organic]

Matt,

Let us write it this way: A general and rigorous proof of Collatz problem is equivalence to the existence of ZF axiom of infinity, which tells us that there must be an inductive set in our model.

Because no axiom can be proved, Collatz problem can't be proved.

[matt grime]

Originally posted by Organic
Matt,

Let us write it this way: A general and rigorous proof of Collatz problem is equivalence to the existence of ZF axiom of infinity, which tells us that there must be an inductive set in our model.

Because no axiom can be proved, Collatz problem can't be proved.

How on earth is Collatz equivalent to the axiom of infinity?

[Organic]

Dear Matt,

Please look at page 1 and 2 in my proof, and see by yourself:

http://www.geocities.com/complementarytheory/3n1proof.pdf

How Collatz is tuned to the internal Binary Tree that stands in tha base of set N.

After you got it, you can see that Collatz problem form this point of view is equivalence to: "if n exists then n+1 exists"(the ZF axiom of infinity).

[matt grime]

I looked at the proof you offer and it doesn't prove anything that you are claiming. It may require the axiom of infinity, but only in the sense that N is infinite.

All it does is show that given a number there is an odd number greater than it and that under the collatz iteration it gets sent to a larger number again. And?

I really think you should learn about maths before you post stuff like this. Oh, I know you're just going to say that I don't understand your proof, and I should be more accepting, but really that isn't the case. That I can't understand your proof is because it is ill-written and makes completely substantiated deductions. It really would be benefical if you actually understood the axiom of infinity, that merely states the existence of an inductive set in our model is required for it to satisfy the ZF(C) axioms.
It reads as though you think the only proofs are those that check every single case.

Here is another trivial example which shows you are wrong:

let I be the iteration given by I(n)=1 if n is even n+1 if n is odd

given any r = 2^k+1, it is beyond the scope of the binary tree of base k as is r+1, therefore we see that it is undecidable (or contradicts the axiom of infinity, whatever) that repeated iterations of I send all numbers to 1 eventually. Clearly that is a false assertion but the logic is no different than your Collatz deductions.

[Organic]

Matt,

If you write:

All it does is show that given a number there is an odd number greater than it and that under the collatz iteration it gets sent to a larger number again. And?

It means that you still don't understand my proof.

Again, the main point is that I find 1-1 and unto between the internal structure of set N (given by The Von Neumann Hierarchy) and Collatz Binary Tree.

The The Von Neumann Hierarchy Binary Tree is (as much as i know) a new structure that I have found and never used before by anyone.

By this proof I connect between the micro level and the macro level of set N members, and only by this connection between micro and macro we can see that Collatz problem is equivalent to ZF axiom of infinity.

Until know you are still see my proof only from macro level (where there is a meaning to odd or even) but again my proof can be understood only by the connection between this level and set N internal Binary Tree structure (where concepts like odd or even don't exist) that constructed by {} recursion.

[matt grime]

I think the word your searching for is bijection.

1. define the collatz binary tree.
2. define what it means for this tree to be in bijection with the set of natural numbers
3. is there some structure this bijection is preserving.
4. for that matter define THE von neumann heirarchy tree, you have defined trees for each power of two, how does that define it for N? oh i know, that'll be the axiom of infinity induction.
5. explain why this alleged bijection means collatz is false (it can't be undecidable in ZF by your reasoning becuase it contradicts something that is true, hence it must be false).
6. prove it is actually a bijection, whatever it is.
7. if the Collatz conjecture is equivalent to the axiom of infinity, then it is true in ZF, becuase the axiom of infinity is true in ZF. so you're contradicted yourself again and it isn't undecidable (perhaps you should actually look up the meaning of the words you use in a mathematical dictionary). It then won't be true in other set theories without the axiom of infinity. fine, but that's not what you've shown - in fact you are claiming to have proved two contradictory statements. there are results that are true depednding on the set theory you use - Shelah has some nice results on this and EXT groups infact. If you have shown it to be equivalent to the axiom of infinity then well done, you've proved it's true in ZF, and you didn't even realize it. Things like that make me wonder even more about your level of mathematical knowledge.


And why didn't my counter example pass whatever criteria you have for these things? It is defined in the same manner as Collatz, so if my example doesn't make sense on the macro level because odd and even have no meaning, then neither does the Collatz iteration.

[matt grime]

Shall we presume that the COllatz tree is the one that has base node labelled by 1, from this is a branch to a node labelled by 2, from that to a branch labelled by 4, from that a branch to a node labelled by 8, there will then be one node out of 8 to 16, from 16 two branches to nodes labelled 5 and 32, and so on, each node labelled r having one or two branches out of it to nodes labelled 2r and, if applicable (r-1)/3.

You claim this is in bijection (as a tree, or a toplogical space, or what?) to the infinite bi-leaved tree which I think is what you mean by the von Neumann tree. Clearly that is not an isomoprhism of trees. as the base node for one has only 1 branch out of it, and there are two branches out of the base node in the neumann case.


If the collatz tree has only one component with labels for all the naturals, then collatz is true, by the way. but then you seem to think the collatz conjecture is undecidable in ZF and equivalent to the axiom of infinty....

[Organic]

Dear Matt,

I made a real major change in my proof, please read it and tell me what do you think, thank you.

http://www.geocities.com/complementarytheory/3n1proof.pdf

[matt grime]

Several things spring to mind

1. How and why have you assigned the numbers 1 to 5 with the labelvon neumann heirargy above them, what on earth are they?


2. the phrase any n in collatz sequence. erm, what'd the Collatz sequence in this sense. n is a number, putting it in bijective correspondence with {} which you usually indicate is the empty set is an odd thing to do

3. the stuff about transfers is meaningless.

4. How is a sequence equivalent to an axiom? Do you mean that the statement 'the COllatz conjecture is true' is equivalent to the axiom of infinity? in that case YOU HAVE PROVED THE COLLATZ conjecture is true in ZF.

[Organic]

Hi Matt,


1. How and why have you assigned the numbers 1 to 5 with the labelvon neumann heirargy above them, what on earth are they?

Each sub branch of the micro Binary Tree is the part that gives to each n member its unique cardinality (please see page 1).

To each sub brunch in the macro Binary Tree, there is an equivalent sub branch in the micro Binary Tree.

Therefore each sub branch in the macro Binary tree can be 'labeled'
by the macro Binary Tree 'labels', as we clearly shoe in page 2.

2. the phrase any n in collatz sequence. erm, what'd the Collatz sequence in this sense. n is a number, putting it in bijective correspondence with {} which you usually indicate is the empty set is an odd thing to do

Any 'labeled' leaf in the macro Binary Tree, can be put in a bijection with {}, which is the internal atom of each {} self recursion that exists in the micro Binary Tree.

Please compare beteen pages 1 and 2.

3. the stuff about transfers is meaningless.

Only if you don't understand 1 and 2.

This is the interesting part of my proof, which using the bijection between macro-micro Binary tree, to show how each transfer in Collatz sequence always leading us to the scope of another n’s internal structure.

Please compare macro tree in page 1 to the micro tree in page 2.

4. How is a sequence equivalent to an axiom? Do you mean that the statement 'the COllatz conjecture is true' is equivalent to the axiom of infinity?

Because there is a bijection between micro and macro tree, and we don’t care about the direction of each transfer, then each transfer is least n to n+1.

Therefore we can conclude that Collatz sequence is equivalent to the ZF axiom of infinity, which is the axiom that defines N members.

in that case YOU HAVE PROVED THE COLLATZ conjecture is true in ZF.

Again, Collatz conjecture is true in ZF, but cannot be proved within ZF, exactly as any axiom is true within its own system, but by definition cannot be proved within its own system.

For me any axiomatic system has the potential to be a part of more general axiomatic system, and in this case each axiom of the included axiomatic system become a theorem, therefore can be proved in the more general axiomatic system.

[matt grime]

I don't think those are the labels I was referring to - I meant the numbers 1,2,3,4,5 that appear in blue red green brown and pink resp. They only belong to a specific binary tree for 2^k, they would have to change for each k, and what are they there for anyway?


You've not defined micro and macro binary tree.

Is THE von Neumann tree the one of infinte depth?

It would appear 'transfer' is what you are misusing for iterate.

Or perhaps transfer is what you mean by the action of the bijection. Please clearly define this bijection, tell me its domain and range.

Actually look up some maths about mappings, becuase if micro and macro trees (whatever they are) have a bijection between them then they are the same.

Try writing words, Organic and explaining what you mean. Don't say a picture's worth a thousand words again, because it's not in maths, and isn't in real life either.


From what you write about axioms, once more it becomes clear you dont' understand the maths here. If the Collatz conjecturebeing true is equivalent to the axiom of infininty, then it IS true in ZF, but not in another set theory without the axiom of infinity. The only things that are necessarily true in any axiomatic theory are those things that are deducible from the axioms (or if its negation contradicts them). Thus it is entirely possible for some results to be true or false depending on the set theory you use.

Why can't you understand that? It is necessary for consistency.

Besides, it seems like it's time for yet another reiteration that the axiom of infinity does not create the integers. The natural numbers exist, they are the counting numbers, ok? Now thanks to Russell we found out that juist naively labelling all classes of objects sets leads to some serious problems, he got round it with types, but after many years of thought Zermelo-Frankel came up with axioms that sets must satisfy in order to make sense and agree with what we want to be true in the main. So, these classes of objects exist already, what ZF does is tell us which form sets, and when we can do things on them that we can do with sets.
We want the natural numbers to form a set in our theory, hence we need the axiom of infinity to tell us that an infinite set necessarily exists in our model (I'm using the term model loosely) - if it weren't there then we could take the sets to only be those with finite cardinalities (I think)) and we wouldn't have forced the Naturals to be included - they would still exist, but we couldn't call them a set.

So, let's take a simple parallel - take the theory of commutative rings, there are axioms there to satisfy in order for things to be called rings. The axioms there don't force us to have multiplicative inverses, so we can't use field theory here unless we explicitly squash any use of mult inverses.

We can add another axiom requiring multiplicative inverses, and then we have only fields in there, anything true in the old ring theory is still true here.

So, if something is equivalent to the axiom of infinity, then it IS true in ZF. I know I'm just repeating myself but it's something that is so obvious I can't put it in any other terms. You're just saying that Collatiz is equivalent to the assertion the natural numbers constitue a set, which is true. Perhaps you think it is only by assumption, in some sense you are right, but more illustratively it is true by definition - we want the Naturals to be a set, if they weren't what kind of maths would that be? Anything that can be deduced from the axioms of a theory is true in that theory - if it is directly equivalent to an axiom then the proof is very short. You don't have a proof of that though - what you have falls well short of the standards required to be called rigorous. You do maths by demonstrating the conjecture is equivalent to something you know to be true. There is some base level of things which are true automatically - the axiom of infinity is one of them. If you want to do maths your way, then no result can be proved, because nothing is true in your system. It's like trying to write a novel but not allowing yourself to use any words or letters.

[Organic]

Hi Matt,

You do maths by demonstrating the conjecture is equivalent to something you know to be true. There is some base level of things which are true automatically - the axiom of infinity is one of them. If you want to do maths your way, then no result can be proved, because nothing is true in your system. It's like trying to write a novel but not allowing yourself to use any words or letters.

Only Collatz-like sequences are true by my proof.

No theory of x is x itself therefore any axiom is a potential theorem in a more general theory.

Shortly speaking, any theory is an open information system that can be changed when its paradigms are changed.

Closed systems cannot survive in the long run.

I don't think those are the labels I was referring to - I meant the numbers 1,2,3,4,5 that appear in blue red green brown and pink resp. They only belong to a specific binary tree for 2^k, they would have to change for each k, and what are they there for anyway?

My answer in the previous post is referring exactly to these numbers.

Please check again micro and macro sub branches 'labeled' by the same 'labels' in both pages 2 and 3, and see how they are fitting (overlapping) exactly each other by their basic structure.

Words or diagrams are just tools to express our ideas.

You belong to a community that express its ideas by words, I don't.

In my opinion, there is nothing in the tools themselves that determines if some idea is mathematical or not.

[matt grime]

But youu don't produce any explanation of generating these trees, or what they are!

Please answer these simple question.

1. The diagram labelled collatz sequence macro tree. What is it, how are you generating it, what are the blue dots.

2. Why is the phrase 'any n in Collatz sequence has a 1-1 and onto with {}...' there? It is meaningless in the English langauage an mathematically.


3. You speak of 'this binary tree' that stands at the 'base of N' implying it is unique - you've drawn at least 3 of them, so it isn't unique. What is it?
I asked it it is the infinite bifurcating diagram with two outgoing edges at each vertex, and one incoming (except at the initial vertex where there are no incident edges), you didn't answer.



4. You still speak of 'Collatz sequence' with no article, definite otherwise. What do you mean by this?


5. It is customary in writing a proof of a conjecture (or whatever it is you are now claiming) to state what the conjecture says. Why isn't a statement of what you are trying to prove included?

6. The theory doesn't change; a different theory may replace it, supersede it or just be developed in tandem. If one wants to do anything requiring set theory, then ZF(C) is the current vogue. Ok, this isn't a question.

[matt grime]

In fact, just state what it is you are trying to prove, so that you at least demonstrate you know what the Collatz Conjecture says. That doesn't mean post a link to some Wolfram site, just in plain English explain what it is that you understand the Conjecture to say. Perhaps then we might even see what you mean by 'Collatz sequence'

[Organic]

Matt,

1. The diagram labelled collatz sequence macro tree. What is it, how are you generating it, what are the blue dots.

Don't you see the similarity of these sub branches in maco and micro Binary Trees?

These sub branches defined in the micro tree, but fits to the macro tree, where Collatz sequence (marked by the blue dots) was first defined.

2. Why is the phrase 'any n in Collatz sequence has a 1-1 and onto with {}...' there? It is meaningless in the English langauage an mathematically.

I translated it from Hebrew, any way you can use "bijection" instead.

. You speak of 'this binary tree' that stands at the 'base of N' implying it is unique - you've drawn at least 3 of them, so it isn't unique. What is it?

Where did i draw 3 of them? In the base of the Von Neumann Hierarchy
there is exactly one and only one Binary Tree.

I asked it it is the infinite bifurcating diagram with two outgoing edges at each vertex, and one incoming (except at the initial vertex where there are no incident edges), you didn't answer.

I don't undestand your question.

4. You still speak of 'Collatz sequence' with no article, definite otherwise. What do you mean by this?

I don't understand what you write here too.

5. It is customary in writing a proof of a conjecture (or whatever it is you are now claiming) to state what the conjecture says. Why isn't a statement of what you are trying to prove included?

Thank you, I'll do it tonight.

6. The theory doesn't change; a different theory may replace it, supersede it or just be developed in tandem. If one wants to do anything requiring set theory, then ZF(C) is the current vogue. Ok, this isn't a question.

No you no me can know what exactly happen when some mutation takes place in some real or theoretical system.

[matt grime]

there is one and only one binary tree? ok, what is it?

is it the infinite continuation of the finite depth tree you have drawn under the title collatz sequence macro tree?

that is is it the one that goes

x
0 1
01 01
01 01 01 01
....
.
.


off 'to infinity' where you can't say what x is? you'll have to imagine the branches drawn in, and try and justify it correctly - html is a pain.


As for just saying 'Collatz sequece'

A sequence is a function from some set, S, to N, that is you can think of it as sequence

(a_1,a_2,a_3.....) a_i in S


so what is 'collatz sequence'?

*a* collatz sequence can be defined, i suppose, by setting a_1 to be some integer, r, and obtaining the next term by the collatz iteration. However there is no unique collatz sequence - a different r defines a different sequence, so which one of these are you talking about? or is it something else entirely.

the conjecture states that given any r, the above defined sequence eventually reaches 1.

that will do for now

[Organic]

Matt,

Sorry about my Math language, wich is very poor as you already know.

By sequence I mean the tree of Collatz sequences for example:

http://michael.cleverly.com/funstuff/3x+1/collatz2.jpg


The blue dots in my diagrams are equivalent to this tree.

Any magenta branch in the macro tree goes directly to 1.

[matt grime]

Ok, I'm beginning to see the picture.


1. The binary tree is what i said it is - the infinite thing above. At the k'th level you number the nodes from 1 to 2^k using binary expansions (this means by the way that the only consistent choice for the initial node is 1). You then superimpose those green lines and blue dotsby saying join dot r with dot t iff at least one of the following hold:

r=2t
t=2r
r=3t+1
t=3r+1

ok.

What youve observed is that at any level, it must be that there is a number that is sent outside that level to the next one, for instance you need to construct level k+1 and higher to indicate all th points where 2^k - 1 gets sent/can come from



let's address what you are claiming this implies.

In order to provide a proof of the Collatz conjecture in ZF, one must have all the Natural numbers. However it is in the assumption of the Collatz conjecture that the naturals exist and are a set in ZF. If we are not assuming the axiom of infinity, then the natural numbers do not form a set in the theory we are using, and hence Collatz is undefined in that theory. However, all the axiom of infinity states is that the natural numbers form a set, it does not imply the Collatz conjecture s true. The truth of the Collatz conjecture is not therefore equivalent to the axiom of infinity.

We must presume the axiom of infinity is in our set theory to define the Collatz conjecture, that is all.

the naturals exist, whether or not you chose to call them a set and do so independently of the set theory you use, there is nothing in what you've written to suggest that the assumption the naturals form a set is equivalent to Collatz being true or not.


I think the key here is your views on axiomatic set theory and the natural numbers, nothing to do with Collatz. The axiom of infinity just states that the naturals are an (inductive) set in ZF(C).

[Organic]

Matt,

You are close so maybe in this post you will understand my proof.

Please pay attention to the fact that only Collatz-like sequences take you always from some n to another n in the micro tree.

If you combine all these "movements" to one direction (let us choose the positive direction) then by collatz-like sequences you get the "If n exists then (at least) n+1 exists" which is equivalent to ZF axiom of infinity.

[matt grime]

But, Organic, the Collatz Iteration is defined on N already, that is all the n's in N 'exist' already. You're finding problems where there aren't any, and in particular you still seem to think that the axiom of infinity is what makes N exist, when that isn't what it states. The axioms of a group say do not force idnetity elements to exist, do they?

[Organic]

Matt,

Can N exists without the axiom of infinity?

Please show me how you can say that there are inifinitly many n's in N without the axiom of infinity?

[matt grime]

yes, they do exist without the axiom of infinity. This is what I've been attempting to explain since the year dot (English phrase meaning from the very beginning) all the axioms of *a* set theory do is tell you what collections of things constitute sets. It as an attempt to put the theory of sets onto a firm footing so that we can avoid things like the Russell paradox - the collection of all sets that do not contain themselves 'exists' but it is not a set (in ZFC). If you take all the other axioms of ZF(C) apart from the axiom of infinity, then there is no way to force the Natural numbers to be a set - that is it is not possible to deduce the existence of an infinite set from the other axioms alone.

There is some philosophical debate as to what one means by 'exist', but the fact that the greeks were able to talk about the natural numbers (and the rationals etc) without knowing the axiom of infinity should tell you something! The existence philosophy boils down to 'is there some higher plane where '1' genuinely exists in the same way that the computer in front of you exists, or is it that we force it to exist in this universe and it is purely an invention of this universe' this is the so-called idea of the platonic realm. Actually that wasn't a very good example, but if you want to learn about it try looking up the word 'qualia' which is the idea that oneness is some higher concept independent of 1, and that 1 is just some realization of it. But please don't think you can use that to do anytihng with 'infinity' with because, as we established earlier, the idea that 'infinity' exists is a myth, infinite is just 'not finite' which has associated implications, and useful properties.

[matt grime]

Just seen the edit.


Suppose that there are only finitely many natural numbers ( a natural number is one obtained by adding 1 repeatedly), then let M be the largest. M+1 is a natural number by definition, and strictly larger than M. This isn't the axiom of infinity. Don't like that one?

Clearly 2 is a natural number, so M>1 if it exists, yet then M*M > M # so M can't exist.

The axiom of infinity means that there is an infinite set, that is all.

[Organic]

Matt,

By saying: "yes, they do exist without the axiom of infinity."

You maybe use a vary deep platonic realm.

Well, I do not accept your platonic point of view, because without the axiom of infinity all you have is the empty set by the empty set axiom.

[matt grime]

No, they exist by induction, they form a set in ZF by the axiom of infinity. If you want a set theoretic proof for these things try the peano postulates.

I suppose saying they exist inductively sounds wrong doesn't it?


We can count, we know what 1 dog is, 2 dogs, etc, the counting things are natural numbers, we get them by adding 1 repeatedly. Now the question is can we form the SET of natural numbers? Well, the collection of natural numbers exists, and is clearly infinite, that they are a set in ZF is useful, and exactly the reason why the axiom of infinity is there, in ZF they for an inductive set. That's all, I'm not appealing to any platonic realm, only that we can count and have labels for this counting. Are you getting this yet? That it is whether or not we can call it a set, that's all? You can define classes of arbitrary things, they just don't turn out to behave as sets morally ought to if our logic is to be consistent. Set theory is just a formalization of what we know ought to be true, in a way that makes it consistent - the class of sets that do not contain themselves is not a set, yet it still exists.

I knew I'd regret telling you abuot the platonic realm. You already seem to have misunderstood what was admittedly a bad explanation of it; the explainer must take the blame.

[Organic]

Peano postulates and ZF axoim of infinity are the same induction.

[matt grime]

Peano just puts the naturals on a well founded set theory footing. Induction is there, it is independent of the Axiom of infinity, which after all requires induction, to demonstrate that there is actually an infinite set.


The natural numbers exist, that they form a set is all we are saying with the axiom of infinity.

[Organic]

Without this induction you can have one and only one element.

This induction does not define the internal structure made by Von Neumann.

My proof combines between Von Neumann recursive method and the ZF induction.

By this recursive(micro)-inductive(macro) point of view we can check the invariant symmetry that exists in the base of the examined elements.

[matt grime]

Originally posted by Organic
Matt,

By saying: "yes, they do exist without the axiom of infinity."

You maybe use a vary deep platonic realm.

Well, I do not accept your platonic point of view, because without the axiom of infinity all you have is the empty set by the empty set axiom.

But how can the empty set exist? By your logic it can't. Because the existence of the empty set in your logic requires the axiom of the empty set, which requires the empty set to exist... oh no, your circular reasoning goes horribly wrong! Have you read anything that talks about set theory at all, ever? Get a book from the library, you might learn something. And when you understand it, try again.

[matt grime]

OF course if you don't like that, then how about this (which i noted earlier but you ignored)

if one wants to prove that Collatz iterations of all natural numbers reach one, then you must be presuming that the natural numbers all already exist to even define the Conjecture, so there can be no issue in presuming they exist to prove it. Anyway, you haven't proved anything, well apart from that you don't know what you're talking about, yet still feel confident enough to tell me I'm wrong about everything.

[Organic]

Nothing circular here.

The axiom of the empty set and the empty set are the same.

The axiom of infinity use this axiomatic existence as its input and using induction to define its pruducts.

Von Neumann Hierarchy uses the same axiomatic existence as its input and using recursion to define its products.

My proof combines between Von Neumann recursive method and the ZF induction.

By this recursive(micro)-inductive(macro) point of view we can check the invariant symmetry that exists in the base of the examined elements.

[matt grime]

You think the axiom of the empty set is the empty set? Wow, that's a new one. You know you really ought to learn some maths, Organic if you're going to persist in this.

Erm, there is no such thing as ZF induction as far as I'm aware. Please post some reference for this.

Inducion is a mathematical technique, using the existence of the 'inductive' set that the axiom of infinity requires, one can construct an infinite set. And? How does this bear any relation to anything here?

At least accept that the existence of the Natural numbers is implicit in the statement of the Collatz conjecture. Therefore assuming their existence in the proof is acceptable!


That you do not seem to understand the ideas of set theory can be left alone - go and get a book and learn about them. The axiom of infinity does not state the naturals exist, it states that there must be an infinite set.

[Organic]

Let me put it this way, the ZF axiom of the empty set uses x
where x is something and then it says that for any x(=something),
x(=something) not in some set X.

This is nothing but a negative and complicated way to say positively and simply: "there exist set X with no elements" notated as {}.

ZF induction is my mistake, it has to be the ZF axiom of infinity induction.

You don't see the invariant symmetry that exists in both Von Neumann
recursion(micro level) and Collatz sequences (macro level).

The ZF axiom of infinity products (if n exists then n+1 exists implies N members) are in the level of the macro tree, where Collatz sequences exists, therefore we cannot distinguish between them and the Collatz sequences because in the macro level they are the same elements.

Eech n in the induction macro level has its own internal unique structure produced by Von Neumann recursion.


When we examine the invariant Binary tree that stands in the basis of both Collatz sequences and Von Neumann recursion, then and only then we can see that Collatz sequences and ZF axiom of infinity are the same iteration.

[matt grime]

Oh. My. God. For the god knows how many'th time, what is the ZF axiom of infinity induction?. You are the only person I have ever known use it, Google for the phrase. You and responses to you were the only hits the last time I did it.

There is the axiom of infinity which states there must be a set W satisfying a certain property: if y is in W then y u {y} is in W. There is absolutely no mention of induction there. Now, using induction we can conclude that there is a set which does not have finite cardinality. That's it. It is not the ZF axiom of infinity induction.

For some reason you seem to think that it is the axioms of ZF that force things to exist. Exercise: just using the axioms, how do you see that the class of all continuous Real valued functions on a compact hausdorff topological space exists?

The class exists, independently of ZFC. Which is just a formal attempt to put the theory of sets onto a firm mathematical footing. These things all exist albeit in some mathematical sense anyway.

Read a book on set theory, please, I'm begging you.

[Organic]

Matt,

Don't you see that "if y is in W then y u {y} is in W" is exactly the same as "if n is in N then n+1 is in N"?

"if n is in N then n+1 is in N" is an induction.

Please look at the other things that I wrote in the previous post.

Thank you.

[matt grime]

You are missing the subtle point about what the word induction means here, in particular it is properly the principle of mathematical induction which states that, suppose P(r) is a set of statements indexed by N. Then if P(n) => P(n+1) and P(1) is true, it follows P(r) is true for all r in N. What ever short hand you may have for it, and whatever the superficial similarity, the axiom of infinity that if y is in W, then yu{y} is in W is not the principle of mathematical induction. Using the principle of mathematical induction one concludes that the axiom of infinity states there is a set whose cardinality is not finite. Roughly speaking the set theory one uses must have an inductive set. These are subtle issues, but you're lack of precision with mathematical language causes you to say things that are wrong, often very wrong - you have said repeatedly that a set is equal to an inequality for instance. This is a common error you commit - saying that two different things are equal or the same.

In particular if n is in N then n+1 is in N is an 'inductive step' in a proof by induction.

Yes there are similarities, and that is how the axiom of infinity comes to be in the form it is - there are equivalent formulations (inductive sets).

Anyway, enough of this. Back to the main point.

Your proof. what stage is at at now. It is on at least its third rewrite isn't it. Each time you insist it is a proof, and each time you have to rewrite it cos you realisze you've made another mistake, and each time you're as certain that it is a proof. How about relabelling it as an idea cos you don't seem to understand what a proof is.

[Organic]

1) I rewrote it because i saw that you don't understand it.

2)"if n is in N then n+1 is in N" is not an 'inductive step' but an induction forced by an axiom (the ZF axiom of infinity).

3) Please this time read what is below and write your remarks:

Let me put it this way, the ZF axiom of the empty set uses x
where x is something and then it says that for any x(=something),
x(=something) not in some set X.

This is nothing but a negative and complicated way to say positively and simply: "there exist set X with no elements" notated as {}.

If you don't see the invariant symmetry that exists in both Von Neumann recursion(micro level) and Collatz sequences (macro level) then you can't understand my proof.

The ZF axiom of infinity products (if n exists then n+1 exists implies N members) are in the level of the macro tree, where Collatz sequences exists, therefore we cannot distinguish between them and the Collatz sequences because in the macro level they are the same elements.

Eech n in the induction macro level has its own internal unique structure produced by Von Neumann recursion.


When we examine the invariant Binary tree that stands in the basis of both Collatz sequences and Von Neumann recursion, then and only then we can see that Collatz sequences and ZF axiom of infinity are the same iteration.

[matt grime]

OK, lets ignore the infinity stuff, cos it is implicit in the definition of the Conjecture that we are proving it for all n in N.

Right, now you put the collatz tree in some sort of bijection with the binary tree. However, as each integer n occurs an infinite number times in the labelling on the binary tee, how are you deciding which one you want?

Seeing as you tend not to see things like this, here's a proof of that assertion. let r be fixed, the number 2^r-1 occurs on the k'th level for every k > r. So which of those are you picking?

I say some sort of bijection cos I don't see what is in bijection with what clearly. what are the two sets that are in bijection?

[Organic]

The Binary Tree is in bijection with itself, or if you like macro tree (the induction level) is in bijection with the micro tree (the internal recursion level).

[matt grime]

So, something is in bijection with itself? Stop presses. You claimed something about the n being in bijection with {} or something, please repost that.

[Organic]

Please overlap the example of number 4 in page 1(the internal recursion leval)
with the example of page 2 (the induction level).

http://www.geocities.com/complementarytheory/3n1proof.pdf

Dear Matt, I'm going to sleep. have a very good night, and thank you very very much.

[Hurkyl]

Some comments for all:

You belong to a community that express its ideas by words, I don't.

In my opinion, there is nothing in the tools themselves that determines if some idea is mathematical or not.

It's not the tools, it's how you use them. Mathematics can be done with pictures, but you have to use them the same way you use words and formulas; by following rules of deduction that let you deduce new pictures from old pictures.

The tricky thing is that pictoral arguments say absolutely nothing about things that get described by words and formulas unless you provide a mapping between picture world and word/formula world.

Sometimes when I've done "proofs" via deductions of pictures, I was only looking for results in picture-land and I was happy. Other times, I was working in word/formula-land and I carfeully constructed a picture-land that had a correspondence back to word/formula land so that my picture proofs really do translate into facts in word/formula-land.


In order to provide a proof of the Collatz conjecture in ZF, one must have all the Natural numbers. However it is in the assumption of the Collatz conjecture that the naturals exist and are a set in ZF.

It takes a bit of effort, but you can state the Collatz conjecture in ZF without ever mentioning N, thus the Collatz conjecture can be defined even if you deny the existance of N.

Of course, it does require you to define the term "natural number", but that can be done without the axiom of infinity. You just don't get the convenience of having a set of all natural numbers.


"If n exists then (at least) n+1 exists" which is equivalent to ZF axiom of infinity.

The axiom of infinity states (accepting your slight abuse of notation):

"There is a set, S, such that if n is in S, then n + 1 is in S".

Please try to understand the difference.


The statement "If n exists then (at least) n+1 exists" would be better said as "For any n, there exists a set we write as n+1". This fact is guaranteed by the axioms of the sum set and the axiom of the pair set:

Given n:
The axiom of the pair set proves the set {n} exists.
The axiom of the pair set proves the set {n, {n}} exists.
The axiom of the sum set proves that n U {n} exists
and "n U {n}" is what is really meant in set-theory land, not "n + 1"



Can N exists without the axiom of infinity?

The answer to this statement (interpreted mathematically) is yes.

ZF is an incomplete theory; to the best of my knowledge, the statement "N exists" is independant from the axioms of ZF without the axiom of infinity.

Thus N can exist without the axiom of infinity.


Please show me how you can say that there are inifinitly many n's in N without the axiom of infinity?

Without the axiom of infinity, I can prove that there does not exist a finite set that contains all of the natural numbers. Here's the sketch:

Suppose F is a set of all natural numbers. The function f(n) = n + 1 is a 1-1 mapping from F onto a proper subset of itself. Therefore F is infinite.

Therefore, no finite set can contain all natural numbers.


Peano postulates and ZF axoim of infinity are the same induction.

Nothing in the Peano postulates say anything about sets. ZF does nothing but postulate the existance of a set with a given property. Induction is a mathematical technique for proof and thus cannot be one or more axioms. None of the three things mentioned in your sentence can be the same.


Without this induction you can have one and only one element.

Tell me how many sets you want proved exist and I'll do it for you without any form of induction. I'll do two sets for you, as a freebie.

{} exists by the axiom of the empty set.
{{}} exists by the axiom of the pair set.

{} is unequal to {{}} because {} is in {{}} but {} is not in {}.

Thus, I've proven two sets, {} and {{}} exist.


The axiom of the empty set and the empty set are the same.

The axioms of a mathematical theory are not the objects of that mathematical theory.

[matt grime]

And I thought I was the only one dumb/obstinate enough to try and explain stuff to Organic. I don't suppose by any chance you're a set theorist, are you Hurkyl? I'm one of those representation theory people and attempting to explain ZF and set theory is not something I'm particularly comfortable with - my statements here are reasonably informed opinions, but I'm absolutely convinced I've stated things that would have the genuine set theorist spitting blood. The use of 'model' throughout is not in the sense of model theory (or if it is it was a lucky guess).

[Hurkyl]

And I thought I was the only one dumb/obstinate enough to try and explain stuff to Organic.

I've spent probably a whole year trying to explain things to Organic. I'm on a break. [:)]


I don't suppose by any chance you're a set theorist, are you Hurkyl?

I don't really know enough about anything, yet, to have earned any sort of specialist title!

[Organic]

Hurkyl and Matt,


"There is a set, S, such that if n is in S, then n + 1 is in S".


Nothing in the Peano postulates say anything about sets. ZF does nothing but postulate the existance of a set with a given property. Induction is a mathematical technique for proof and thus cannot be one or more axioms. None of the three things mentioned in your sentence can be the same.

Before we can speak about a concept like set we have to define its existence.

For example:

ZF axiom of the empty set

There is a set, A, such that for any x, x not in A.



Also before we can use Peano postulates we have to define some "minimal input" to start with.

For example:

Peano's first postulate

1 is in N (also can be understood as: 1 is a natural number)



In both axiomatic systems, there are at least two basic concepts, which are:

1. A container (in the above examples they are A, N)
2. A content (in the above examples they are x, 1)



Now let us examine ZF axiom of infinity and Peano's second postulate:

ZF: There is a set, S, such that if n is in S, then n + 1 is in S.

Peano: If n is in N, then its "successor" n' is in N.

As we can see, both axioms are forcing induction on some content, which means that they are the same axiom.

The result of this forced induction is infinitely many elements called “the natural numbers”.

These infinitely many elements cannot be defined without the forced induction axiom.



Without the axiom of infinity, I can prove that there does not exist a finite set that contains all of the natural numbers. Here's the sketch:

Suppose F is a set of all natural numbers. The function f(n) = n + 1 is a 1-1 mapping from F onto a proper subset of itself. Therefore F is infinite.

Therefore, no finite set can contain all natural numbers.


Isn't f(n) = n + 1 equivalent to the axiom of infinity?




Tell me how many sets you want proved exist and I'll do it for you without any form of induction. I'll do two sets for you, as a freebie.

{} exists by the axiom of the empty set.
{{}} exists by the axiom of the pair set.

{} is unequal to {{}} because {} is in {{}} but {} is not in {}.

Thus, I've proven two sets, {} and {{}} exist.


Thank you Hurkyl for correcting my mistake here.


The axiom of the empty set and the empty set are the same.


In this address (http://www.cs.bilkent.edu.tr/~akman/jour-papers/air/node5.html ) I have found this:

“The Null Set Axiom guarantees that there is a set with no elements, i.e., the empty set {}. This is the only set whose existence is explicitly stated".

Again, let me put it this way, the ZF axiom of the empty set uses x
where x is something and then it says that for any x(=something),
x(=something) not in some set X.

This is nothing but an indirect and complicated way of what we can say directly and simply: "there exist set X with no elements" notated as {}.

(In my opinion any element whose existence is explicitly stated, is equivalent to the axiom that defines it).

"The Pair Set Axiom states the existence of a set which has a member when the only existing set is {}. So the set {{}} can now be formed now and we have two objects {} and {{}}. The application of the axiom repetitively yields any finite number of sets, each with only one or two elements.

It is the Sum Set Axiom which states the existence of sets containing any finite number of elements by defining the union of already existing sets. Thus
{{},{{}}}U{{{},{{}}}} = {{},{{}},{{},{{}}}}

However it should be noted that all these sets will be finite because only finitely many sets can be formed by applying Pair Set and Sum Set finitely many times.

It is the Axiom of Infinity which states the existence of at least one infinite set, from which other infinite sets can be formed. The set which the axiom asserts to exist is {{},{{}},{{},{{}}},{{},{{}},{{},{{}}}}}.

Thus, the ZF universe simply starts with the {} and extends to infinity. It can be noticed that cumulative hierarchy produces all finite sets and many infinite ones, but it does not produce all infinite sets.”


Well Pair Set Axiom and Sum Set Axiom are the basis of Von Neumann recursion.

If you don't see the invariant symmetry that exists in both Von Neumann recursion(micro level) and Collatz sequences(macro level) then you can't understand my proof.

The ZF axiom of infinity products (if n exists then n+1 exists implies infinitely many N members) are in the level of the macro tree, where Collatz sequences exists, therefore we cannot distinguish between them and the Collatz sequences because in the macro level they are the same elements.

Eech n in the induction macro level has its own internal unique structure produced by Von Neumann recursion.


When we examine the invariant Binary tree that stands in the basis of both Collatz sequences and Von Neumann recursion, then and only then we can see that Collatz sequences and ZF axiom of infinity are the same iteration.

General: any iteration that is based on root or exponent value 2, is equivalent to the ZF axiom of infinity iteration.

Please look again at: http://www.geocities.com/complementarytheory/3n1proof.pdf

Thank you,

Organic

[matt grime]

You seem to not be able to distinguish between different objects.

A function is not an axiom, a set and an axiom, etc.

You also appear to have invented another new term: ZF axiom of infinity products.

I don't really have the inclination to even think about what you mean in your new view of it.

[matt grime]

quote:
In my opinion any element whose existence is explicitly stated, is equivalent to the axiom that defines it

I missed that one. that's probably one of the more silly things you've said, and boy is that a hotly contested fight.

so many counter-examples and corrections spring to minf, but what's the point? You keep insisting you are correct and have this proof of something, and you keep on having to correct it and each time you come back just as vehement that we are wrong and you are right.... hell some times you don't even correct it, yet still claim it is correct despite the evidence to the contrary (I wonder if you are still claiming a set is equal to an inequality...?)

What is it you are claiming anyway - that the Collatz conjecture is undecidable? Do you take this from the assertion that COnway proved some Collatz like conjectures are undecidable? did you even look at his proof? do you even know what undecidable actually means? Seeing as number theory uses ZF, and you claim that he Collatz conjecture is EQUIVALENT to the axiom of infinity, then you've probed Collatz is true (in ZF).

You haven't done that though.

Tell you what, you produce a nice picture that, say, proves the 4-subspace problem is wild, and maybe we'll take you a little more seriously. Until then I'll just keep pointing out the gross abuses of the English langauge and mathematics you are disguising your lack of ability in.

[Hurkyl]

Before we can speak about a concept like set we have to define its existence.

Why would you think that?

If I have a list of properties that something called a "set" obeys, I don't need to know anything about the existance of a "set" in order to reason about those properties.

(This is a familiar idea even in "everyday" logic; we often call it a "hypothetical scenario" in such a context)


In both axiomatic systems, there are at least two basic concepts, which are:

1. A container (in the above examples they are A, N)

There is no container concept in peano arithmetic. (There are also quite a few other theories that have no container concept)

The peano axioms give a list of properties for three undefined terms: the unary predicate "is a natural number", the constant symbol "1", and the unary function "successor".

"Natural number" is no more a container than "human" is; in laymen's terms, "Natural number" is just a name we give to a 'type' of thing; it is a 'category' (I mean this in the lay sense, not the technical sense).


An aside on notation: sometimes, we like to use notation that looks like set notation, even when there are no sets involved. To use a different example, we might choose to write the logical predicate "Z is an ordinal number" in the form Z \in \mathrm{OR}, even though there is not (and in ZF, cannot be) a set OR that contains all ordinal numbers.


ZF: There is a set, S, such that if n is in S, then n + 1 is in S.

Peano: If n is in N, then its "successor" n' is in N.

As we can see, both axioms are forcing induction on some content, which means that they are the same axiom.

First off, we have to assume that you intend the phrases "n + 1" and n \cup \{ n \} to be synonymous.

Secondly, how can they be the same axiom?

The axiom of infinity guarantees the existance of a set, but I can't use the Peano axiom "If n is a natural number, then the successor of n is a natural number" to prove the existance of any set.


And, incidentally, the set guaranteed by the axiom of infinity cannot be proven to be an inductive set under the "n + 1" operation.



Isn't f(n) = n + 1 equivalent to the axiom of infinity?

As per a later statement you make, I would like to point out that the existance of the function I used here is given by the axiom of the power set and the axiom of subsets.

The axiom of power sets guarantees the existance of the set P(P(F)). (the power set of the power set of F)

P(P(F)) contains, among other things, all ordered pairs (a, b) where a and b are elements of F.

The axiom of subsets guarantees the existance of a set, call it f, which is the subset of P(P(F)) that contains only ordered pairs of the form (x, x + 1). This is the function we would more commonally write f(n) = n + 1.

More precisely:


f := \{ x | x \in \mathcal{P}(\mathcal{P}(F)) \wedge(\exists a \in F : x = (a, a+1) \}




(In my opinion any element whose existence is explicitly stated, is equivalent to the axiom that defines it).

My limited comprehension of your ideas may be at fault here, but this sounds awfully like confusing "x-content" with an "x-model".



Finally, I would like to make a comment on models.

I mentioned earlier that:

The peano axioms give a list of properties for three undefined terms: the unary predicate "is a natural number", the constant symbol "0", and the unary function "successor".

(I changed "1" to "0" because I prefer formulations of the natural numbers that include zero)

Allow me to emphasize that these are undefined terms of the theory of peano arithmetic. They are NOT "n is in N", "{}", and "n U {n}".


When we make these identifications, we are making what mathematicians call "a model of the natural numbers in ZF". A model, more or less, consists of taking the undefined terms of one theory and defining them in another theory. There are additional details, such as you have to verify that all of the axioms of the one theory are satisfied when you define the terms in the other theory.

[Organic]

If I have a list of properties that something called a "set" obeys, I don't need to know anything about the existance of a "set" in order to reason about those properties.

(This is a familiar idea even in "everyday" logic; we often call it a "hypothetical scenario" in such a context)

Any x in some theory cannot be but a model(X), therefore we always have to be aware to the combination of model() and X, where model() is the global state and X is some local state.

Model() is the container (global state).

X is the content. (some local state).

x is the product of model(X) (the combination or relations between global and local states).

Shortly speaking any theory is first of all model() or if you like an empty container (a global state) "waiting" to some X (some local state) to be its examined concept.

From this point of view, any theory must be aware to the relations between the global and the local, otherwise it cannot use its full potential.

If Peano Axioms is a theory then first of all is a model()(a container) that "needs" some X(a content) to deal with.

For example:

1 is in N (also can be understood as: 1 is a natural number)

x=model(X) where x cannot be but a combination of model()(= theory of numbers) and X(= the concept of a number).

It means that if we want to understand x(=1 in this case) we have to "put on the table" its combination (container-content) property.

Shortly speaking, 1 cannot be but 1 in N, whether you say it or not.

Therefore There is a container concept in Peano arithmetic.

Again, there can be a big problem for us to understand and develop deeper connections between so called different areas of research, if we don't take in account the global-local or container-content relations.

What I wrote here also can explain why ZF axiom of infinity and Peano first axiom are the same axiom.

And this axiom can be called "The forced-induction axiom".


My limited comprehension of your ideas may be at fault here, but this sounds awfully like confusing "x-content" with an "x-model".

All products of some theory are nothing but an x-model, an axiom is a product therefore an x-model.

I am talking about the hierarchy of dependency among these products.


The basic level is the axioms level, and on top of it there is the hierarchy of products, which can exist iff they do not contradict the axioms level.

But when we need an axiom that directly determines the existence of some element, it means that there is no hierarchy here but "the same lady with a different dress".

[matt grime]

Originally posted by Organic
x=model(X) where x=1 cannot be but a combination of model()(= theory of numbers) and X(= the concept of a number).

Therefore any product in any theory cannot be but a combination of container and content.

The first thing in there translates as

1= theoryofnumbers(conceptofanumber)

which is beyond words.

[Organic]

No it is simple.

x=thory_of_some_concept(THE_CONCEPT).

thory_of_some_concept() is the container.

THE_CONCEPT is the content.

x is the container-content relations.

[matt grime]

Er, I think you missed the joke again. you say concept_of_A_number, well which one is 1 the realization of the concept of. I was pointing out the imprecisoin of your terminology again.

[Organic]

x=model() or x=CONCEPT.

In both cases x cannot be a product of some useful system.

Therefore x has its meaning only by theory(=model() )and X(= some concept) relations.

Concept_of_a_number is not a Theory_of_numbers, and theory_of_numbers without Concept_of_a_number is an empty model(=model() ).

By the way x=model(X) is not under Russell's Paradox, because the global level(=container) transcendent any local level(=content).

Er, I think you missed the joke again. you say concept_of_A_number, well which one is 1 the realization of the concept of. I was pointing out the imprecisoin of your terminology again.

Concept_of _a_number is the intuitive ability to count things without any supported theory.

[matt grime]

but surely 2=model(conceptof anumber) as well, and thus 1=2. Stop using equals when you don't mean equals. In fact, stop using extant mathematical terms for your own bizarre and ill-informed asinine concepts full stop.

[Organic]

The '=' in x=model(X) is not an equal sign, but like in a cumputer program where x gets as input the result of model(X) output.

[matt grime]

I wouldn't know whether to laugh or cry if I cared anymore

[Organic]

What is you problem know?

I used a computer program syntax to show that any useful product of some theoretical system cannot be but a model(CONCEPT) combination, as I explained in my last posts.

[matt grime]

I've been reading your crap for the last couple of months and telling you repeatedly that you are not using mathematical terms correctly (and last time I checked this was a maths forum, not a computer science one) and after all that time only now do you tell me that you're using the equals sign to not mean equal? You don't find that slightly amusing?

[Organic]

Most of the time I used the words 'input' and 'output' when a wrote x=model(X) expresion.

For example: http://www.geocities.com/complementarytheory/Theory.pdf

Instead of using titles, all you have is to open your mind and try to understand what infront of your eyes, and not in what is the title of the forum, because sometimes the difference between areas is not so clear.

[Hurkyl]

Warning: variable X is used without having been initialized.

[matt grime]

As with the vast majority of your output you fail to define what you use. You cannot expect people to second guess you. But this is worse than second guess you, becuase you are horribly misusing extant terms in your own private theory without explaining what the **** is going on. You cannot expect people to understand you if you do this, nor care to read what you write. If your material is difficult to read because it is poorly written you will get no interest. Your standard of presentation is poor enough as it is without deliberately obfuscating the issue in inappropriate notation.

[Organic]

No Dear Hurkyl,

This is the whole point. an actual concept does not have to be initialized.

Only x in the expression x=model(X) has to be initialized.

[Organic]

Matt,

Only in the expression x=model(X) I used '=' as an input sign, so again what is the problem?

Please show me another part of my work where '=' is unclear.

Thank you.

[Hurkyl]

Only x in the expression x=model(X) has to be initialized.

Try telling a C compiler that. [;)]


If you are using '=' as assignment (which, btw, is generally frowned upon in Computer Science; using a symbol like \leftarrow is generally preferred), then any previous value of 'x' is irrelevant because it's being assigned the return value of model(X). However, since X has not been initialized, (IOW you haven't given us any indication whatsoever what X is supposed to be) model(X) is an entirely meaningless expression.

Incidentally,

error: no definition found for function model.


P.S. whoops, I didn't notice that this was put in the math forum! Back to TD you go

[Organic]

Well Hurkyl,

I see that you did not read the part where I write exactly What is X and why it is not have to be initialized.

The reason is very simple, X represents a constant in x <-- model(X) expression.


Also please can you answer to what is written below?

Thank you.


If I have a list of properties that something called a "set" obeys, I don't need to know anything about the existance of a "set" in order to reason about those properties.

(This is a familiar idea even in "everyday" logic; we often call it a "hypothetical scenario" in such a context)

Any x in some theory cannot be but a model(X), therefore we always have to be aware to the combination of model() and X, where model() is the global state and X is some local state.

Model() is the container (global state).

X is the content. (some local state).

x is the product of model(X) (the combination or relations between global and local states).

Shortly speaking any theory is first of all model() or if you like an empty container (a global state) "waiting" to some X (some local state) to be its examined concept.

From this point of view, any theory must be aware to the relations between the global and the local, otherwise it cannot use its full potential.

If Peano Axioms is a theory then first of all is a model()(a container) that "needs" some X(a content) to deal with.

For example:

1 is in N (also can be understood as: 1 is a natural number)

x <-- model(X) where x cannot be but a combination of model()(= theory of numbers) and X(= the concept of a number).

Concept_of _a_number is the intuitive ability to count things without any supported theory.

Concept_of_a_number is not a Theory_of_numbers, and theory_of_numbers without Concept_of_a_number is an empty model(=model() ).

It means that if we want to understand x we have to "put on the table" its combination (container-content) property.

Shortly speaking, 1 cannot be but 1 in N, whether you say it or not.

Therefore There is a container concept in Peano arithmetic.

Again, there can be a big problem for us to understand and develop deeper connections between so called different areas of research, if we don't take in account the global-local or container-content relations.

What I wrote here also can explain why ZF axiom of infinity and Peano first axiom are the same axiom.

And this axiom can be called "The forced-induction axiom".


My limited comprehension of your ideas may be at fault here, but this sounds awfully like confusing "x-content" with an "x-model".

All products of some theory are nothing but an x-model, an axiom is a product therefore an x-model.

I am talking about the hierarchy of dependency among these products.


The basic level is the axioms level, and on top of it there is the hierarchy of products, which can exist iff they do not contradict the axioms level.

But when we need an axiom that directly determines the existence of some element, it means that there is no hierarchy here but "the same lady with a different dress".

By the way x <-- model(X) is not under Russell's Paradox, because the global level(=container) transcendent any local level(=content).

[matt grime]

Let us take some other theory: group theory. The axiom of identity does not 'create' the identity. groups exist whether or not we have a theory of groups. How did Galois manage to use them since the modern axiomatic form of group theory wasn't developed until around 70 years after his death.?

[Organic]

Dear Matt,


Maybe I am a living example of concepts without (yet) a rigorous framework as their theory.

This is exactly the meaning of x <-- model(X) expression.

By this expression we can understand that any useful x is at least a combination of some initial concept X(some constant) and model or a theory where we can develop it.

Galois case is a beautiful example that supports x <-- model(X) where Galois' work is the initial X and modern axiomatic form of group theory is model(X), and its useful product is x <-- model(X).

Be aware that I am not talking about the existence of X, but on the useful existence of x as a product of some model() and X combinations.

I made major corrections is this paper:

http://www.geocities.com/complementarytheory/Theory.pdf

Pleae write your remarks on it.


Thank you.


Yours,


Organic

[matt grime]

At no point do you even define what these xs and Xs are. They could be goats, cars, or George Bush's DUI criminal record. What is 'actual infinity'? Oh, I know we're supposed to read Riemann's Balls pdf aren't we. Where you don't define it either. What's a set, Organic? What are any of these things?

The first two sentences of you reply above are just rubbish that makes no sense in the English language. The meaning of your postion on x<--model(X) is EXACTLY that you are a living ecample of concepts without a rigorous framework as their theory? Eh? You are a mathematical ill-defined concept?

What are you rambling on about with Galois? group theory is now a refinement of the ideas of galois and a generalization of the key points in his work. it is nothing to do with models or anything.

So go back and explain everything you write, explain the meaning of all the terms you misuse. I for one am no longer going to read your musings as evidently learning maths is not something you want to do.

[Organic]

The first two sentences of you reply above are just rubbish that makes no sense in the English language. The meaning of your postion on x<--model(X) is EXACTLY that you are a living ecample of concepts without a rigorous framework as their theory? Eh? You are a mathematical ill-defined concept?


You missed the point, concepts without a theory is only X.



Another example:

By x <-- model(Taniyama-Shimura) we mean that x is a product a more comprehensive theory which is based on the deep relations between modular forms and elliptic curves.

[Hurkyl]

X represents a constant

If X was a constant, you would be able to say exactly what it is, or specify a list of properties that characterize the constant symbol 'X'.

However, you later state:

some initial concept X(some constant)

...

Which seems to indicate that X isn't a fixed thing in your theories; it can be different things depending on the situation. We call that a variable.




Anyways, let's work through an example. Take this formal system. (I won't call it a theory becuase there is likely some disagreement between what mathematicians call a theory and what you call a theory)


In addition to ordinary logic, this formal system has the unary predicates P and L, and the binary predicates I and = (with = written in the usual infix notation) and the following axioms:


\begin{array}{l}
\forall a, b: I(a, b) \implies (P(a) \wedge L(b)) \\

\forall a, b: (P(a) \wedge P(b) \wedge \neg(a = b)) \implies (\exists c: L(c) \wedge I(a, c) \wedge I(b, c)) \\

\forall a, b, c, d: (I(a, c) \wedge I(a, d) \wedge I(b, c) \wedge I(b, d)) \implies (a = b \vee c = d)
\end{array}



So what happens when you apply your ideas to this?

[Organic]

Hurkyl,

X in my theory is a geneal represetation for a constant, for example:

X is 'Taniyama-Shimura theorem'

x <-- model('Taniyama-Shimura theorem')

X is 'The concept of a number'

x <-- model('The concept of a number')

X is '...'

x <-- model('...')

[Hurkyl]

Hurkyl,

X in my theory is a geneal represetation for a constant, for example:

X is 'Taniyama-Shimura theorem'

x <-- model('Taniyama-Shimura theorem')

X is 'The concept of a number'

x <-- model('The concept of a number')

X is '...'

x <-- model('...')

In other words, X is a variable.

[Organic]

Let us say that X is like a dummy variable.

Shortley speaking, in any use of the general form of x <-- model(X),
X place is taken by some constant.

X is only a place holder.

[Hurkyl]

Let us say that X is like a dummy variable.

Shortley speaking, in any use of the general form of x <-- model(X),
X place is taken by some constant.

X is only a place holder.

In other words, X is a variable. (as the term is used in logic)


Anyways, I presented a formal system. How do your ideas apply to it?

[Organic]

If you represent a formal system please show me where are you in this system?

Show some mathematical reseach that take your abilites to develop math language as a legal part of math.

For example look at this paper:
http://www.geocities.com/complementarytheory/count.pdf

[Hurkyl]

I didn't say I represent a formal system, I said I presented a formal system.

[Organic]

Hurkyl,

Please see my answer to some old post of you in this tread.



If I have a list of properties that something called a "set" obeys, I don't need to know anything about the existance of a "set" in order to reason about those properties.

(This is a familiar idea even in "everyday" logic; we often call it a "hypothetical scenario" in such a context)

Any x in some theory cannot be but a model(X), therefore we always have to be aware to the combination of model() and X, where model() is the global state and X is some local state.

Model() is the container (global state).

X is the content. (some local state).

x is the product of model(X) (the combination or relations between global and local states).

Shortly speaking any theory is first of all model() or if you like an empty container (a global state) "waiting" to some X (some local state) to be its examined concept.

From this point of view, any theory must be aware to the relations between the global and the local, otherwise it cannot use its full potential.

If Peano Axioms is a theory then first of all is a model()(a container) that "needs" some X(a content) to deal with.

For example:

1 is in N (also can be understood as: 1 is a natural number)

x <-- model(X) where x cannot be but a combination of model()(= theory of numbers) and X(= the concept of a number).

Concept_of _a_number is the intuitive ability to count things without any supported theory.

Concept_of_a_number is not a Theory_of_numbers, and theory_of_numbers without Concept_of_a_number is an empty model(=model() ).

It means that if we want to understand x we have to "put on the table" its combination (container-content) property.

Shortly speaking, 1 cannot be but 1 in N, whether you say it or not.

Therefore There is a container concept in Peano arithmetic.

Again, there can be a big problem for us to understand and develop deeper connections between so called different areas of research, if we don't take in account the global-local or container-content relations.

What I wrote here also can explain why ZF axiom of infinity and Peano first axiom are the same axiom.

And this axiom can be called "The forced-induction axiom".


My limited comprehension of your ideas may be at fault here, but this sounds awfully like confusing "x-content" with an "x-model".

All products of some theory are nothing but an x-model, an axiom is a product therefore an x-model.

I am talking about the hierarchy of dependency among these products.


The basic level is the axioms level, and on top of it there is the hierarchy of products, which can exist iff they do not contradict the axioms level.

But when we need an axiom that directly determines the existence of some element, it means that there is no hierarchy here but "the same lady with a different dress".

By the way x <-- model(X) is not under Russell's Paradox, because the global level(=container) transcendent any local level(=content).

[Hurkyl]

How do your ideas apply to this formal system?


In addition to ordinary logic, this it has the unary predicates P and L, and the binary predicates I and = (with = written in the usual infix notation) and the following axioms:



\begin{array}{l}

\forall a, b: I(a, b) \implies (P(a) \wedge L(b)) \\


\forall a, b: (P(a) \wedge P(b) \wedge \neg(a = b)) \implies (\exists c: L(c) \wedge I(a, c) \wedge I(b, c)) \\


\forall a, b, c, d: (I(a, c) \wedge I(a, d) \wedge I(b, c) \wedge I(b, d)) \implies (a = b \vee c = d)

\end{array}

[Organic]

Please translate it to plain English.

[Hurkyl]

Let's say P(a) means "a is a P-thing", L(b) means "b is an L-thing", and I(a, b) mean "a and b interact".


\forall a, b: I(a, b) \implies (P(a) \wedge L(b))


Interactions only occur between a P-thing and an L-thing.


\forall a, b: (P(a) \wedge P(b) \wedge \neg(a = b)) \implies (\exists c: L(c) \wedge I(a, c) \wedge I(b, c))


For any two different P-things, there is an L-thing that interacts with both of them.



\forall a, b, c, d: (I(a, c) \wedge I(a, d) \wedge I(b, c) \wedge I(b, d)) \implies (a = b \vee c = d)


If each of two L-things interact with each of two P-things, then the L-things are the same or the P-things are the same.

[Organic]

Please tell me if my conclusions are right.

Axiom 1: There are no self interactions between L L or P P.


Axiom 2 : There must exist this one to two structure


Pa <-->|
|-L
Pb <-->|


Axiom 3 : the first exists by axiom 2

Pa <--> La |(= Axiom 2)
Pb <--> La |

La <--> Pa
Lb <--> Pa

[Hurkyl]

So how does all this x = model(X) stuff fit in?


I'm not sure what your drawing for the third axiom is supposed to mean.

[Organic]

If each of two L-things interact with each of two P-things, then the L-things are the same or the P-things are the same.

Axiom 3 : the first structure already exists by axiom 2

Pa <--> La
Pb <--> La

Pa <-->|
|-L
Pb <-->|

La <--> Pa
Lb <--> Pa

La <-->|
|-P
Lb <-->|

Any way, the least structure of these axioms (as much as I see) cannot
be but one to two structure, which is the building-block of the Binary-Tree, or even more general the interactions between integration (sum) and differentiation (parts).

x is always a combination of model() and X, for example:

model() <-->|
|-x
X <-->|

[Hurkyl]

I still don't know what the whole thing means. What are the structures? What can you do with them? Why are they useful? How do they relate to P-things, L-things, and interactions?

(I admit I'm pretty sure about a partial answer to the last of these questions)


And this is the most confusing of all:


model() <-->|
|-x
X <-->|


How can this make any sense? Are you saying that, for instance, "model()" is a P-thing or an L-thing? How can this be if "model()" is supposed to be some sort of function? And if this is the meaning of "x = model(X)", how can it possibly apply to any other theory (such as set theory)?


(P.S. does it ever make sense, to you anyways, to say "X = model(x)", "y = model(Y)" or "j = model(C)"?)

[matt grime]

you want your detractors to demonstrate they can do maths (I know, but it's a like a habit that's bad for you, reading your latest murdering of mathmatics, and I'm finding it hard to go cold turkey) ok, here are some things i've proven that no one else has published as far as we are aware

"Show some mathematical reseach that take your abilites to develop math language as a legal part of math"

How about this:

Let M be the module category of some finite dimensional group algebra over a (countable) algebraically closed field. Then the modules induced from a subgroup do not necessarily form a definable subcategory (in the sense of Krause). In particular, there are certain groups with normal subgroups of index p (=char of field) with the direct limit of induced modules from the subgroup not induced from any module.

that do you?

If you want I can give you some sufficient conditions on relative stable categories that ensures they are compactly generated. Interested?

[Organic]

Matt,


you want your detractors to demonstrate they can do maths

Another typical example of your emotional response that does not give you the chance to understand what you read.

I did not ask anyone to show me how good mathematician he is, but asked for some legal brach in mathematics that researches our cognition's abilities to create math language.

Also I gave an example for this kind of a research:

http://www.geocities.com/complementarytheory/count.pdf

When you read it then you will understand what I mean.

[Organic]

Dear Hurkyl,


My number one law is: think simple (which is not think trivial).

When we think simple then we get the chance to (sometimes) see the deeper interactions between so called different things.

The 3 axioms that you gave me define the building-block of one-to-many interactions structure.

1) side 1 of this building block is different from side 2 in this building-block by at least 2 properties.

Property 1) P is not L

Property 2) If P is one then L is at least two, and vise versa.

Therefore we can get:


Pa <-->|
|-L
Pb <-->|

OR

La <-->|
|-P
Lb <-->|

But the deep invariant thing is the one_to_many structure.

----|
|--
----|

So (as I see it) our systems are the same in this level.

I take this level, mark my basic elements on it and get:

model() <-->|
|-x
X <-->|

1) x cannot be but a model of X, where X cannot be anything but a thing that can be translated to a model of itself.

Shortly speaking, X-itself can be an x-model of X-itself.

2) x can exists iff there are at least two things: a theory of X(=model() ), X.

3) x is the interaction of model() and X, notated as x <-- model(X).

4) model() is the container, X is the content, therefore x is container-content interactions.

5) Also model() is the global, X is the local, therefore x is global-local interactions, which means that any x can be understood only by its global-local interactions or container-content interactions.

x needs at least two parents to exist, Mama model() and Papa X.

how can it possibly apply to any other theory (such as set theory)?

x cannot be but a container-content interaction where content can be at least nothing XOR something.

Know please read these short papers by this order, and see by yourself some examples that are based on this way of thinking:

1) http://www.geocities.com/complementarytheory/ET.pdf

2) http://www.geocities.com/complementarytheory/AHA.pdf

3) http://www.geocities.com/complementarytheory/Everything.pdf

4) http://www.geocities.com/complementarytheory/ASPIRATING.pdf

5) http://www.geocities.com/complementarytheory/Theory.pdf

6) http://www.geocities.com/complementarytheory/HelpIsNeeded.pdf

[Hurkyl]

I still don't know what "x", "model()", nor "X" are.

The reason I presented a specific formal system for you to analyze was so that you could explicitly write down what "X" is, et cetera.

e.g. you say that X represents some constant. Well, show us the constant. Fill in the blank: "X = _"

[Organic]

For example:

X='INFINITY'


Please look at: http://www.geocities.com/complementarytheory/Theory.pdf

[Hurkyl]

How does "X='INFINITY'" relate to the system I presented?

[Organic]

Infinity has two basic sides:


Potential ----|
|-- Actual
Potential ----|


Please look at:

http://www.geocities.com/complementarytheory/Everything.pdf

http://www.geocities.com/complementarytheory/RiemannsLimits.pdf

[matt grime]

Originally posted by Organic
Matt,


Another typical example of your emotional response that does not give you the chance to understand what you read.

I did not ask anyone to show me how good mathematician he is, but asked for some legal brach in mathematics that researches our cognition's abilities to create math language.

Actually this is what you asked:

"Show some mathematical reseach that take your abilites to develop math language as a legal part of math."

It is a sentence that makes little sense, but my respsonse was one of the possible interpretations.

[matt grime]

Define 'actual infinity', what is it? Seriously?

As a general aside, if my emotional responses do not allow me to understand (mathematically) what I read, how come I've got several qualifications in mathematics?

[Hurkyl]

So is 'actual infinity' a P-thing or an L-thing? And what about 'potential infinity'? And there are two different potential infinities?!?!


P.S. you asserted:

Axiom 1: There are no self interactions between L L or P P.

Which is incorrect. Here is a model that is a counterexample:


The only object in the model is M. M is an L-thing and a P-thing, and M interacts with M.

Formally:

P(M), L(M), I(M, M)

This model satisfies the three axioms of my formal system, yet there is an interaction I(P, Q) where P and Q are both L-things, and where P and Q are both P-things.

[Organic]

Your first axiom: "Interactions only occur between a P-thing and an L-thing."

(P(M) interacts with P(M)) is not (P(M) interacts with L(M))
(L(M) interacts with L(M)) is not (L(M) interacts with P(M))

(P(M) interacts with P(M)) or (L(M) interacts with L(M)) are not
allowed by your first axiom.

Therefore Axiom 1: There are no self interactions between L L or P P.


So is 'actual infinity' a P-thing or an L-thing? And what about 'potential infinity'? And there are two different potential infinities?!?!

By your system I do not care about P-thing or L-thing because I am looking only for the invariant product of your axiomatic system, with is:

----|
|--
----|

and the reason that P-thing or L-thing are not the invariant of your axiomatic system is because there can be:

Pa <-->|
|-L
Pb <-->|

OR

La <-->|
|-P
Lb <-->|

So, the invariant thing is the interaction sctucture itself, which is:

----|
|--
----|

Now, let us look again at this:

Infinity has two basic sides:

Potential ----|
|-- Actual
Potential ----|

Actual infinity is {__} or {} contents (the word many is not allowed).

Potential infinity is {a,b,...} (the word many is allowed).

Again, if you want to understand my point of view, you have no choice but to understand my models here:

http://www.geocities.com/complementarytheory/Theory.pdf

http://www.geocities.com/complementarytheory/Everything.pdf

I'll say it again, if you choose not to read my papers, it is equivalent to:

"I asked you a question but I don't care about your answer".

[Hurkyl]

Your first axiom: "Interactions only occur between a P-thing and an L-thing."

You've made the unjustified assumption that something cannot be both a P-thing and an L-thing; A P-thing can interact with another P-thing if one of them is also an L-thing.


By your system I do not care about P-thing or L-thing because I am looking only for the invariant product of your axiomatic system, with is:

My system doesn't have an invariant product; that's something you created.


I'll say it again, if you choose not to read my papers, it is equivalent to:

"I asked you a question but I don't care about your answer".

The thing is, I'm asking you about horses and you're telling me about mermaids!

You are right, I don't care about your answers because you have not been answering my questions.


All of your attempts to explain the phrase "x = model(X)" have been little (if anything) more than a facy way of saying "It's a mathematical phrase with three parts; 'x', 'X', and 'model()'". You've tried giving examples, but all of your examples are your ideas that the rest of us have been telling you over and over that we don't understand.


My purpose over the last couple threads was to try and create a concrete, yet simple example for you to explain your ideas. In particular, I was hoping for precise definitions of the individual components of "x = model(X)". I purposefully tried to create an example that beared little resemblance to the other examples you have tried to make, in hopes that you would stick to the example instead of replacing it with the concepts which you should know are not sufficiently clear to the rest of us.

Apparently I didn't do a very good job; just about the only thing you've said about my system is an unjustifiable claim that there are no self-interactions between L's or P's. Apart from that, you've managed to rewrite everything else so you can start talking about invariant products and potential vs actual infinity again.


So yes, I don't care about your answer, because it's not an answer to my question.

[matt grime]

when someone asks you a simple direct question they are entitled to a simple direct answer addressing the question. It's as if someone asks you what what the annual level of rainfall in kinshasha is and you say, ah here is a link to the website of the encyclopedia britannica!

[Organic]

Hi Hurkyl,



First, thank you very much for your efforts to give me some bridge to your world.

OK, I see, I looked at P-thing as P AND its content (or at L-thing as L AND its content) , where you looked only on the contents of of each of them, isn't it?.

Any way, let us take inf <-- model('INFINITY') where 'INFINITY' stands for actual infinity that cannot be used as it is by Math.

But the model of it (notated as inf) which i call it a potential infinity, can be used by Math.

Actual infinity is too string or too weak to be used as information in Math language.

The srong limit is marked by {__} content, and the weak limit is marked by {} content (the word many cannot be used).

Potential infinity is marked by {a,b,...} (the word many is used).

Let us start from this model, that shows the difference between actual infinity that is marked by {___}, and potential infinity that is marked by {a,b,c,...}:

http://www.geocities.com/complementarytheory/RiemannsLimits.pdf


To understand better {___} and {} contents please look at:

http://www.geocities.com/complementarytheory/4BPM.pdf

The infinitely long base of the green trianlges is {___} content.

The infinitely long base of the empty trianlges is {} content.


Please look at both models and write your detailed remarks.



Thank you,

Orgainc

[matt grime]

when you write {a,b,...} you are implying that the 'model of infinity' is an infinite countable set of elements. Is that really what you want to do. Why is infinity a set like this? It is infinite, but that is not the same thing.

[Organic]

Matt,

Please show me how you notate an infinitley many elements which are not countable.

By my point of view there is no such a thing "uncountalbe infinitely many elements".

I already proved it here:
http://www.geocities.com/complementarytheory/NewDiagonalView.pdf

[matt grime]

you proved no such thing there because of your repeated misuse of 'the axiom of infinity of induction'.


in that article you claim certain things have certain properties which have been absolutely refuted beyond any doubt by my arguments in various threads on this issue. that you do not understand the refutations is because you do not understand mathematical arguments. At various times, you claim that an infinite list has a 'last' element, that something is a priori countable for no reason that is true in mathematics, you claim results about infinite sets from finite ones despite plentiful counter examples to this principle, you want me to go on?

As in this thread, you claim to prove something (incorrectly) and then immediately claim that in fact the opposite is true. in that you 'prove' by an incorrect method that something is uncountable, then state that it isn't. an educated monkey can see it isn't countable (picking a metaphor not at random). in this thread you claim something is deducible (is equivalent to!) some axiom in ZF and then state it is undecidable in ZF! proving only that you don't know what undecidable means.


simply put I cannot understand anyone who cannot understand there is no bijection between N and its powerset and that therefore there are uncountable sets; it's just a defintion.

that wasn't the point i was making anyway - you say infinty is a set that is implicty enumerable. in what sense is that infinity. it is infinite, that is a different thing entirely. in all of this you have not provided a definition of 'infinity' just some claims about infinite things. that is not the same.


you want a 'notation' for an uncountable infinite set? i'm not sure what that means, but how about R, the real numbers that is easily proven to be uncoantable?

[Hurkyl]

OK, I see, I looked at P-thing as P AND its content (or at L-thing as L AND its content) , where you looked only on the contents of of each of them, isn't it?.

I don't know whether to say "yes" or "no", because I don't know what you mean by "the contents of each of them".

The only things I know about P-things and L-things are the three axioms I've listed.

I -have- been specifically avoiding trying to ascribe any meaning to P-things and L-things; I've been considering them merely as things that obey the three axioms I gave.

My best guess is that I have done the exact opposite of "looking only on the contents of each of them".



Do P-things fit into your "x <- model(X)" equation? Do they get substituted for 'x' or do they get substituted for 'X'? What goes in the other spot?


Please show me how you notate an infinitley many elements which are not countable.

We don't try notate them by enumerating their elements. Examples of uncountable sets are \mathbb{R}, \mathbb{N}^\mathbb{N}, and {x | x is in the interior of triange ABC}.

(where the latter, of course, is in the context of Euclidean geometry, and I have already specified the noncollinear points A, B, and C)

[Organic]

Hurkyl and Matt,

Please look again at this model:

http://www.geocities.com/complementarytheory/RiemannsLimits.pdf

Now, please show me a map between infinitely many intersections
representing R set, and the element notated by oo.

If there is no such a map then you have a simple proof shows that infinity or infinite concept in your system is not well defined.

Shortly speaking, you don't know what are you talking about when you use concepts like infinite or infinity in your system.

Cantor, Dedekind, and each one of you as professional mathematician who continue to use their conceptual mistake about the infinite or infinity (by forcing infinitely many elements or intersections of R set on oo) have no reasonable model to talk about.

By forcing infinitely many elements or intersections of R set on oo all you get is a circular and closed system that running after its own tail, therefore prove meaningless proofs when researching infinite or infinity concepts.

And the reason is very simple:

You are not aware to the limits of your system.

And Matt stop telling me about the difference between infinity and infinite, because in both cases a mapping between infinitely many elements is used by standard Math, see for your self:

http://mathworld.wolfram.com/Infinite.html

http://mathworld.wolfram.com/Infinity.html

Don't try to tell me that what is written in Wolfram is wrong, because
I'll send you immediately to the philosophy forum.


Another "great" example of infinity by standard Math can be found here:

http://mathworld.wolfram.com/PointatInfinity.html

[matt grime]

Originally posted by Organic Now, please show me a map between infinitely many intersections
representing R set, and the element notated by oo.


why? what do you mean by the infinitely many intersectiosn representing R set? why is it important to have this map - there is a trivial one sending everything in the domain to the symbol infinity. it's not clear what you want here, or why.

If there is no such a map then you have a simple proof shows that infinity or infinite concept in your system is not well defined.[quote]

but it is. something is infinite if it is not finite. i agree that infinity is not well defined, it is contextual, in the same way as there are different kinds of multiplication operations on different groups

[quote]Shortly speaking, you don't know what are you talking about when you use concepts like infinite or infinity in your system.

i don't think you understand what we mean by anything in mathematics

Cantor, Dedekind, and each one of you as professional mathematician who continue to use their conceptual mistake about the infinite or infinity (by forcing infinitely many elements or intersections of R set on oo) have no reasonable model to talk about.

what does that sentence mean? force infinitely many elements onto something? intersections of R set on oo? they don't make sense.

By forcing infinitely many elements or intersections of R set on oo all you get is a circular and closed system that running after its own tail, therefore prove meaningless proofs when researching infinite or infinity concepts.

And the reason is very simple:

You are not aware to the limits of your system.

And Matt stop telling me about the difference between infinity and infinite, because in both cases a mapping between infinitely many elements is used by standard Math, see for your self:

http://mathworld.wolfram.com/Infinite.html

http://mathworld.wolfram.com/Infinity.html

Don't try to tell me that what is written in Wolfram is wrong, because
I'll send you immediately to the philosophy forum.


Another "great" example of infinity by standard Math can be found here:

http://mathworld.wolfram.com/PointatInfinity.html [/B]

why must wolfram be correct? infinity as they have it is a useful notion that encapsulates the idea of being 'not finite' and perhaps it isn't they who are wrong but you who does not understand what is writte there?

you are the one misusing (mathematical) langauge and saying infinity is a set of some kind or is {__}.

think for a second and define multiplication. see? probably not.

the symbol infinity is used in a variety of ways, the point at infininty of the Riemann sphere, the sum from 1 to infinity and so on. they all have the common thread of denoting 'not finite', or 'at no finite point'. Why do you insist that there is this ACTUAL INFINITY out there? what is it? please, define it clearly. if you are going to use {__} again try and define that becuase you have not produced a defintion that anyone has accepted or understood.

look on the websites you list. show me where

"mapping between infinitely many elements is used"

is written, or anything approaching it. are you trying to use the idea that a set is infinite iff it is in bijection with a proper subset of itselt? but that doesn't tell you what infinity is does it? people abuse langauge by saying 'there are an infinity' of real numbers, but the key here is that it is a phrase 'infinity of', and it means that there are an infinite number of, it doesn't mean infinity is a set in the way you think it is.

[Hurkyl]

Suppose the real numbers is countable. Choose any enumeration of them.

Create a countable collection of intervals such that the i-th interval contains the i-th real number, and has length 1/2i.

This collection of intervals contains every real number, however, the total length of all of the intervals is 1.

[Organic]

Hurkyl,

This is exactly what R is, a fractal where each part of it is the same in any scale that we choose.

Cantor himself used this invariant self similarty upon scales to define R, or what is called sometimes "Cantor set":

http://mathworld.wolfram.com/CantorSet.html

Cantor set is nothing but a Binary-Tree.

Please look here:

http://www.geocities.com/complementarytheory/LIM.pdf

As you can see Cantor set exists in the open interval ({},{__}),
Therefore R cannot use the model of a line.

Shortly sparking the "real line" (a collection of infinitely many objects that construct a one solid element) is a conceptual mistake of modern mathematics, and any result or research that is based on it is nothing but a waste of time.

You can use any collection of nice symbols that you want, but there is nothing but nonsense behind them.

Please read my paper about the CH problem:

http://www.geocities.com/complementarytheory/CL-CH.pdf

[Organic]

Matt,

The Math you use (when dealing with the non-finite) is valuable like a point at infinity.


you are the one misusing (mathematical) langauge and saying infinity is a set of some kind or is {__}.

think for a second and define multiplication. see? probably not.


1) There is no an objective thing like Mathematical language which is disconnected form the people who create it, so there is no use to repeat again on this false thing.

2) For Multiplication please read this:

http://www.geocities.com/complementarytheory/ASPIRATING.pdf

[matt grime]

Originally posted by Organic This is exactly what R is, a fractal where each part of it is the same in any scale that we choose.

Cantor himself used this invariant self similarty upon scales to define R, or what is called sometimes "Cantor set":

that is not the definition of the cantor set; R is not a cantor set. A cantor set is a pefect hausdorf compact totally dsiconnected etc subset of the real line and is unique up to homeomorphism

Cantor set is nothing but a Binary-Tree.

no it isn't. a binary tree does not a priori come with a topology, but giving it one won't work because it is clearly never going to be totally disconnected and perfect etc

Shortly sparking the "real line" (a collection of infinitely many objects that construct a one solid element) is a conceptual mistake of modern mathematics, and any result or research that is based on it is nothing but a waste of time.

You can use any collection of nice symbols that you want, but there is nothing but nonsense behind them.

irony isn't dead!

[Organic]

Cantor set has (by standard Math) the power of the continuum.

Therefore |R|=2^aleph0.

It is easy to show that 2^aleph0 is Cantor set where Cantor set is a Binary Tree:

?
__________________________________

1 0
_____________ _____________

1 0 1 0
_____ _____ _____ _____

1 0 1 0 1 0 1 0
__ __ __ __ __ __ __ __


You know Matt, it is amazing to see how the educational system took your
Independent way of thinking and shaped it to its faceless uniformed shape
which is full of second hand bombastic names that sometimes there is nothing
behind them.

This system killed any flexibility and curiosity that has to be natural parts
of a good researcher, and it did it so good until you can't see simple things
that are standing in front of your eyes.

My heart with you because I think this is a real tragedy.

[matt grime]

the thing you draw isn't even the binary tree in you own paper, it isn't a tree - which are the leaves, the vetices, nodes, whatever? or at least it isn't a tree in anything other than a trivial way.

2^{aleph-0} is a cardinality, it isn't a set, why do you say things are the same when they aren't?

do you know what any of the words compact hausdorff disconnected mean?

thanks for your sympathy. the educational system has completely killed my research abilities, which is why i've done a phd (in maths); yes, your logic is faultless. what it has inculcated in me is a dislike of undefined and therefore unprovable assertions.

[Organic]

the thing you draw isn't even the binary tree in you own paper

As I said my heart is with you.

Please look at this:

http://www.mathacademy.com/pr/prime/articles/cantset/

[Organic]

?
__________________________________

1 0
_____________ _____________

1 0 1 0
_____ _____ _____ _____

1 0 1 0 1 0 1 0
__ __ __ __ __ __ __ __

Is exactly this:
http://www.geocities.com/complementarytheory/PTree.pdf

and this:
http://www.geocities.com/complementarytheory/3n1proof.pdf

[matt grime]

Originally posted by Organic
As I said my heart is with you.

Please look at this:

http://www.mathacademy.com/pr/prime/articles/cantset/

thank you for yet another pointless post, i know perfectly well what a cantor set is, i also know about graph theory. i'm sorry that you don't bother to look up any thing you use until too late, but the tree you draw in your own paper is the infinite bifurcating diagram (infinite in the sense of the number of leaves)and isn't a cantor set - it is connected for instance. as it must be, a tree has the property that any two nodes are connected by a unique path. or didn't you know that? oh look once more your ignorance leads to a problem in the mathematics.

[Organic]

No, the minimal building-block of a Binary tree is simultaneously in two complementary states, which are integration and differentiation.

For eample:

?
|
/ \
/ \
/ \
| |
1 0


And also Cantor set:

?
__________________________________

1 0
_____________ _____________

1 0 1 0
_____ _____ _____ _____

1 0 1 0 1 0 1 0
__ __ __ __ __ __ __ __

[ahrkron]

Originally posted by Organic
No, the minimal building-block of a Binary tree is simultaneously in two complementary states, which are integration and differentiation.


Organic, you are blurring and confusing concepts. It seems that you have a lot of conceptual problems with basic math.

[Organic]

ahrkron,

Please give me an example.

[ahrkron]

I just did. Integration and differentiation have nothing to do with binary trees.

You can probably use both graphs and calculus to represent aspects of some specific problem, but the two concepts are independent of each other, and it is just false that

"the minimal building-block of a Binary tree is simultaneously in two complementary states, which are integration and differentiation."

Also, this statement shows that you are doing an incorrect use of math terminology. Integration and differentiation are operations, not states.

[Organic]

When integration and differentiation complement each other they become states of a structure, which I call the building-block of the Binary-Tree.

Because I used the word "simultaneously" their opposite operational property can be described also as states.

[ahrkron]

Originally posted by Organic
When integration and differentiation complement each other they become states of a structure

No, they don't. The fact that this statement is wrong may pass unnoticed in an informal conversation, but it definitely cannot be used as the basis for the definition of anything in math.

, which I call the building-block of the Binary-Tree.

You cannot "call" things as you please, because you cannot make sure that everybody understands that you are not talking the same language. Your use of words already used in math to designate other concepts can confuse people trying to learn math.

Because I used the word "simultaneously" their opposite operational property can be described also as states.

Again, this is a very informal way to express your ideas. You need to pay much more attention to the accuracy of your statements if you insist in working on math problems.

Just to make it clear: the problem is NOT your command of English, but the lack of precision of your assertions.

[matt grime]

Well done for proving you don't know what a tree is. I asked you about that repeatedly and i thought we established the tree in your article was a genuine tree - the infinite bileaved tree you draw. now we find out you don't know what's going on again. why do you insist on knowing more about maths than the rest of us when you can't even define a tree correctly?

[Organic]

Matt,

This is theory development forum, where I can define a tree in my way.

[Organic]

ahrkron,

No, they don't. The fact that this statement is wrong may pass unnoticed in an informal conversation, but it definitely cannot be used as the basis for the definition of anything in math.

Please look at my paper:
http://www.geocities.com/complementarytheory/ET.pdf

[matt grime]

Originally posted by Organic
Matt,

This is theory development forum, where I can define a tree in my way.

but you didn't define tree though. i asked you about it repeatedly but you never acutally described it properly. undoubtedly it made sense in your head, but you didn't explain it to anyone else. in fact anyone who actually looked in your article would see that you drew a tree as is understood in graph theory, albeit with an infinite vertex set. now you claim the 'tree' is the cantor set. yet the tree is the natural numbers, therefore you're talking crap again! as even allowing for your inconsistent notation you've said it is countable and uncountable, an impossible dichotomy.

[Organic]

Matt,

You don't remember the last version of my work on Cantor's diagonal methed, so please look at it again:

http://www.geocities.com/complementarytheory/NewDiagonalView.pdf

And this time please read all of it, if you want to understand it.

[matt grime]

ah lovely, that heap of garbage again. still using the axiom of infinity of induction despite there being no such thing, still claiming the number of rows is 2^aleph-0 because of the finite case. still wrong despite the number of revisions you've undertaken.

there is not justification for claiming there are 2^aleph-0 rows. there aren't. you are wrong and there really is no simpler way of saying this. so why are there 2^aleph-0 rows? go on pleae state here and now in mathematical terms why there are 2^aleph-0 rows which are enumerable.


here is the counter proof to your assertion

the list you produce is enumerable and is alleged to be the power set of N. Let z be in the power set of N. it is in the list at some point, n(z). by construction though the element at n(z) has only finitely many non-zero entries, therefore as z was arbitrary we have a contradiction.

you've still not managed to refute that counter example to your unfounded assertion.

[Hurkyl]

Allow me to reemphasize my conclusion:

If we assume the real numbers are countable, we can find a set whose total length is 1, yet this set contains every point of the entire real line!


Are you actually comfortable with the implication that the entire real line a length no greater than 1?

[Organic]

Matt,


by construction though the element at n(z) has only finitely many non-zero entries

By what construction?

Please give a detailed example of this construction.

[Organic]

Hurkyl,

That’s exactly the Idea, only a solid line (which means no points in it) has length 1.

No collection of infinitely many points can use the model of a solid line.

Fullness = Solid line = {__} content = Mathematics language strong limit.

For better understanding please look once more at(please pay attention to the Continuity that stands in the basis of empty or full(green) triangles):

http://www.geocities.com/complementarytheory/4BPM.pdf


Emptiness = {} content = Mathematics language weak limit.


Mathematics language is already aware to {} content.


It is the time to fulfill the symmetry by being aware to {__} content.

[matt grime]

Originally posted by Organic
Matt,


By what construction?

Please give a detailed example of this construction. ]]

by your construction. column 1 goes 010101010...

column two goes 001100110011...


column 3 goes 000111000111...

you cycles based on 2^n remember.


therefore the n'th column starts with n zeros. Te resulting infinte matrix you write down is thus strictly lower tringular - just look at the first few diagrams you've drawn in that newdiagonal.pdf

for any row, row r say, reading right to left, all the entries become 0 after the r'th place (if not sooner), thus there are only finitely many non-zero terms in the r'th row (at most r of them).

I've told you this on at least 5 occasions and you've never managed to disprove it. you can't because it's clearly true.

In fact the thing you constructed precisely enumerates the 'finite' power set - the set of finite subsets of N which is countable.

[Hurkyl]

That’s exactly the Idea, only a solid line (which means no points in it) has length 1.

But, by definition of length, [0, 2] has length 2. And [0, 2] is part of the real line, so the real line has to have length no less than 2.


And 1 isn't special; allow me to modify my proof a little:

Suppose the real numbers is countable. Choose any enumeration of them.

Create a countable collection of intervals such that the i-th interval contains the i-th real number, and has length 1/2^(i+1).

This collection of intervals contains every real number, however, the total length of all of the intervals is 1/2.

So now I've proven the length of the real line is no greater than 1/2.


(In fact, I can prove the length of the real line is equal to zero, with an addition to this argument)

[Organic]

Matt,

for any row, row r say, reading right to left, all the entries become 0 after the r'th place (if not sooner), thus there are only finitely many non-zero terms in the r'th row (at most r of them).

There is no r'th place where after it you know exactly what is the next notation (depends on the base value for example: in base 2 the notation can be 0 XOR 1, in base 3 the notation can be 0 XOR 1 XOR 2, in base 4 the notation can be 0 XOR 1 XOR 2 XOR 3, and so on).

Which means that when we dealing with fractalic(=a^b) subsets where b is non-finite, probability enters to the picture and can't be ignored, as i clearly show here:

http://www.geocities.com/complementarytheory/PTree.pdf

This is one of the fundamental mistakes that Cantor did when he researched infinity, and he made this mistake because in his time information theory and fundamental concepts like redundancy and uncertainty were not “must have” concepts of infinity research.

[matt grime]

Your argument doesn't hold water - you fixed base 2. that is how you constructed this object, this 'list' of strings of 0s and 1s. To say that you might have 2s or 3s in the strings of 0s and 1s is frankly misleading, if not a down right attempt to change the subject away from something where you are wrong. You're off again on an unrelated topic.

Jus look at the construction YOU gave, the t'th column starts with t zeroes! You can cleary see that on the r'th row, every entry after the r'th column must be zero. Look at you'ure own diagram where you can see the pattern that all the numbers above the diagonal are 0 - it is trivial to show that this pattern continues in the 'list' as I've just proven.

Can we make it a bit clearer? the t'th entry in row r is the entry from column t, if t>r (and r is fixed remember) then as the t'th column starts with t zeroes and r<t it must be that the r'th entry in that column is 0 becuase all the entries from 1,2,...r,..,t are zero. (This is your construction, yet you do not even understand this simple observation.) So after the r'th place in row r all the entries are zero. thus the r'th row has only a finite number of non-zero entries, (at most r). Thus the corresponding element in the power set is a finite set.

[Organic]

Hurkyl,

This is exactly the beautiful thing in {_} content, for example:

.__. = Finite line = [__]

__ = Infinite line = (__)

.__ = Infinite line = [__)

|{.__.}| = 1

|{.____.}| = 1

|{.________.}| = 1

|{.__}| = 1

|{__}| = 1

Shortly speaking, __ is the essence of an invareant self similarity over scales.

[Organic]

Matt,

To say that you might have 2s or 3s in the strings of 0s and 1s is frankly misleading,

I don't believe that you as a mathematician say such things.

In base 2 the notations are 0,1

In base 3 the notations are 0,1,2

In base 4 the notations are 0,1,2,3

And so on ... (the fixed base value is only in your head)

In my model the important thing is the power_value, where base value
can be any finite natural number.

Because you can't understand the idea of probability in my system, let us look at it without using probability.


by construction though the element at n(z) has only finitely many non-zero entries

Each row is a unique combination of infinitely long sequence of notations (depends on base value) no more no less.

If power value is non finite then “left” side can be (when base value = 2) …000…, …111… or any sub-combination of 01 notations.

[matt grime]

you fixed base 2, don't start changing it now. these things you write down are the indicator functions of sets and thus only take values 0 and 1. Where did you start to bring base 3 or different? what would base 3 even mean. don't answer that as it isn't important and just lets you wriggle out of justifying anything again.

look at your own paper the colums are labelled with powers of 2, that's the key thing.

and as the specific case of 2 here is false, the general case, whatever that might be, cannot be true.

by construction you do not produce any strings with infinitely many 1s in them if you are claiming to be able to enumerate them as you do.

Pick any element in the list, look we've proved it has only a finite number of non-zero entries if your claimed enumeration is true.

therefore your claimed enumeration is false, not mathematics.

and moreover the claimed enumeration sends a string to the binary expansion it denotes, and thus can only be defined for strings with finitely many 1s in.


how is it bad mathematics to state a string of 0s and 1s cannot have 3s or 4s in it? i'd have thought that was bloody well obvious.

[ahrkron]

Originally posted by Organic
Please look at my paper:
http://www.geocities.com/complementarytheory/ET.pdf

I read it.

There is nothing there that applies to the current discussion.

However, there are many comments I can make about its contents.

1. You define "equation trees" in a barely acceptable way (though *very* informal), but then you start using undefined notations and properties. Examples: "symmetry-degree", "information's clarity-degree", the "xor" notation in page 2, the parentheses notation on page 3.

2. On page 3, you say that these trees may be used to "construct and explore complex relations between elements". With "elements", do you mean natural numbers? If so, your assertion is wrong, since natural numbers are well defined already. Your "equation trees" are based on them, and cannot produce further understanding of the numbers they are buit upon!

3. The diagrams on pages 4 and 5 are simple exercises in combinatorics. No new info is gained from looking at them explicitly.

4. On page 6, the product of ET's is never defined. Furthermore, in order to decide if the system is commutative or not, you need to define: a) how to obtain the product, and b) when ET's are equal (the drawings you show there are indeed different, but they could perfectly be two different "symbols" for the same trees).

5. Diagrams in page seven, again, are never defined. They seem entertaining to make, but they also seem useless. The "operations" marked in blue on them also use an undefined notation.

6. Page 9: You never defined what "information point" is supposed to mean.

7. The drawings that occupy the entirety of pages 9 and 10 give no new information, neither do they illustrate any idea from the text. They seem, again, as simple exercises on combinatorics (... related to an unspecified problem).

I'm sorry to say this, Doron, but your "complementarytheory" is really far from providing any insight into natural numbers, and has nothing to do with the issue at hand.

However, you are putting a lot of effort into this. Why not taking a "vacation" from defending your theories and spend the time learning (with an open mind) how things are done in "standard" math? You could gain much from it. I mean it, I'm not trying to patronize you or be condescending.

[matt grime]

in addition to all that, ahrkron, it has come to light that when he says tree he doesn't mean tree as you or i would understand it, but actually a cantor set. apparently it's ok to do this because we're in theory development. as this thread was started by Organic in a math forum and as he never offered a definition of his 'tree', I don't find that a remotely compelling argument, don't know about anyone else. There's also the issue that post facto redefining extant terms to suit yourself is frowned upon: how do we even know that what organic means by cantor set is what the rest of the world means? He didn't even know what a bijection was until it was explained to him, but that hadn't stopped him talking about them before, so the chances of him knowing what a cantor set really is are quite slim.

[Organic]

ahrkron,

All you demonstrate is that at this point you don't understand my work, no less no more.

By the way, because you are a professional mathematician with maybe a lot of knowledge please show me some mathematical branch where multiplication and addition are complementary operations, for example:
http://www.geocities.com/complementarytheory/ASPIRATING.pdf

You are invited to visit my web site: http://www.geocities.com/complementarytheory/CATpage.html

And maybe if you will let your self to be opened to another point of view on Math language, we will be able to communicate with each other in the near future.

Yours,

Organic

[matt grime]

Originally posted by Organic By the way, because you are a professional mathematician with maybe a lot of knowledge please show me some mathematical branch where multiplication and addition are complementary operations, for example:
http://www.geocities.com/complementarytheory/ASPIRATING.pdf

if i go to that article will it define what it means for two binary operations to be complementary? Let's see...

oh look no it doesn't! so simply here and now define what it means for two binary operations to be complementary. Acutally, could you even justify why they are operations on N? each element of N is a collection of partitions of a set with n elements. you mulitply 2*3 and get _some_ of the partitions for 6, but you don't acutally get a number do you, you don't get an element of N. You get a subset of an element of N. How can that define a binary operation from NxN to N? so it is a map from NxN to something that isn't N, even with your definitions of numbers.

I can see an easy way of coming up with some analogy using operads and turning them into, perhaps, some groupoid in a second way, which is after all all we need to do -- your multiplication isn't defined on all partitions, only some of them, at least that is what you appear to say (that bit about structure on page 2)

[ahrkron]

Originally posted by Organic
All you demonstrate is that at this point you don't understand my work, no less no more.

I understand your desire to create on math, and some of your intent while defining the ET's, but I also understand clearly that there are many flaws on your attempt.

It is not necessary to understand the totality of your work in order to see the problems I pointed out in my previous post.

And maybe if you will let your self to be opened to another point of view on Math language

I am very much open to new math concepts and notations. I frequently use quite bizarre notations myself, but it is important to do it in a self consistent way.

Also, if you are to claim that you have uncovered anything about natural numbers, you need to make sure that you translate properly between your own symbols and those used by other people. Otherwise, you need to say "I have discovered fascinating properties about what I call natNumis"; you will surely have less of an audience that way.

Finally, if you allow me, I'd like to advise you not to disregard criticism made on your ideas by just saying "you didn't understand me", since then you will not gain anything from the interaction, and you risk loosing an oportunity to correct perceived or real errors on your work. Very often, good science is done by giving up on ideas that seemed beautiful and that were close to our hearts, but were incorrect nonetheless.

[Edit: added last 4 words]

[Organic]

Dear Matt,

0,1 (base 2) is just some example that can be translated to any fractal which is based on some finite n>1.

A fractal, as you know has a non-linear property and in the case of base 2 we gat a Binary tree, that can be represented in non-compressed way by an ordered matrix of aleph0 width on 2^aleph0 length 01 notations.

[Organic]

Dear ahrkron

Finally, if you allow me, I'd like to advise you not to disregard criticism made on your ideas by just saying "you didn't understand me", since then you will not gain anything from the interaction, and you risk loosing an oportunity to correct perceived or real errors on your work.

First thank you for your gentle and posivite attitude, but from criticism I have have learned that you simply don't understand my work,therefore the detailed remarks that you gave in the previous posts, cannot halp not to you an not to me, at this stage.

If you agree let me start step by step and ask you again this question:

Because you are a professional mathematician with maybe a lot of knowledge please show me some mathematical branch where multiplication and addition are complementary operations, for example:
http://www.geocities.com/complementarytheory/ASPIRATING.pdf

[matt grime]

but it is you assertion that the set rows simultaneously countable, uncountable, and obtained by some undefined induction that is not true.

forget fractals, forget probability.

you cannot, have not, and evidently will not prove anything about the rows. I have, on the other hand, proved you are wrong and you haven't managed to contradict the proof I've offered. Perhaps because you yourself do not even know how to produce this array 'inductively'?

Do you agree that the first column (on the right) is the sequence 01010101... that the second column is 00110011.... etc

that is that by construction there rows are countable - their are as many rows as there entries in a sequence and the sequence is indexed by N.

Now why do you insist that there must be 2^aleph-0 of them when my proof demonstrates that under this construction there are no infinite subsets of N in the construction?

[Organic]

Matt,

Why you repeat yourself?

You proved nothing.

My matrix of 01 notations is:

<---width=aleph0 ---0
|
|
|
|
|
|
|
|
length=2^aleph0
|
|
|
|
|
|
|
V

and both width and length are countable and any row or column is a non-finite unique sequence of 01 notations.

By the way it is constructed the result cannot be but an aleph0 x 2^aleph0 01 matrix, where aleph0 cannot be beyond the infinitly many notations from one hand, and also to be their cardinal on the other hand.

The reason is very simple and can be demostrated in this model:

http://www.geocities.com/complementarytheory/RiemannsLimits.pdf

Therefore the the idea of the transfinite universes is a conceptual mistake.

Your "proof" that takes any ... as ...000... simply demonstrates that you closed under this conceptual mistake.

[matt grime]

I repeat myself because I am correct. We have agreed on how to construct this infinite array, yes?

the first column is on the right, then going to the left we number the columns accordingly.

the first column is 'based on 2^1' and goes 0101010101....

the second colummn reads, downwards, 00110011....

and so on for each column.

the r^th column starts with r zeroes.
then r ones then r zeroes...
Yes?

Let t be any row, the entry in the s'th place (reading right to left) is the entry from the s'th column.

whenever s >t this entry must be zero.

therefore any row is eventually all zeroes, and every element you enumerate in the power set is finite.

Read that argument carefully and don't dismiss it simply because it is me. as you keep telling people they must be open to new ideas, well, so must you.

so go through it and at each point tell me what you think is wrong with the deduction there.

[Organic]

Matt,

I have no choice bu to say it again,

By the way it is constructed the result cannot be but an aleph0 x 2^aleph0 01 matrix, where aleph0 cannot be beyond the infinitly many notations from one hand, and also to be their cardinal on the other hand.

The reason is very simple and can be demostrated in this model:

http://www.geocities.com/complementarytheory/RiemannsLimits.pdf

Therefore the the idea of the transfinite universes is a conceptual mistake.

Your "proof" that takes any ... as ...000... simply demonstrates that you closed under this conceptual mistake.

Please try carefully to understand my model of infinity.

[matt grime]

which .... am i taking to be alll zero? I'm not doing any such thing.

so take my proof point by point and say what's wrong at each point.

anyone else reading this care to tell me where it goes wrong?

(the link you point to is of no importance here)


does the rth column not go 000..01111..11100..0001111.111....

where the are r zeroes and r ones in each block?

look at the diagrams you've drawn. look at the 1st row. doesn't it go 00000 all the way across (i'm reading the numbers backwards here)

doesn't the second row go 1000000...

doesn't the third go 0100000... thence all zeroes?

doesn't the 4th go 1010000000...thence all zeroes

look at the 5th row


take the 6th entry from the right. that is the 5th entry in the 6th column, the 6th column starts with 6 zeroes.

look at the 7th entry in the 5th row, it comes from the 5th entry in the 7th column, the 7th colum starts with 7 zeroes

look at the rth column where r is bigger than 5. it starts with r zeroes, so the 5th one must be a zero.


now i can repeat that for any row, and see that the t'th row is all zeroes after the t'th entry by the way you've defined the columns.

now which step is wrong in that?

now, you claim that the rows enumerate the power set of N, by the standard indicator function argument, I've just shown you every row has a finite number of 1's in it.



here's a little test. Let S_n be a set with n elements. is the union over n in N countable? yes. suppose now S_n has 2^n elements in it. is the union over n in N countable? yes, but you seem to think it isn't because of this axiom of infinity misconception you have lying around.

anyway, take the proof rewritten there including an illustration of the method for the specific exanples of t=1,2..5 and explain where it is wrong.

[Organic]

Matt,

Also please look at this:

3 2 1 0
2 2 2 2
^ ^ ^ ^
| | | |
v v v v
...1 1 1 1 <--> 1
...1 1 1 0 <--> 2
...1 1 0 1 <--> 3
...1 1 0 0 <--> 4
...1 0 1 1 <--> 5
...1 0 1 0 <--> 6
...1 0 0 1 <--> 7
...1 0 0 0 <--> 8
...0 1 1 1 <--> 9
...0 1 1 0 <--> 10
...0 1 0 1 <--> 11
...0 1 0 0 <--> 12
...0 0 1 1 <--> 13
...0 0 1 0 <--> 14
...0 0 0 1 <--> 15
...0 0 0 0 <--> 16
...

[matt grime]

and how are we defining this this time? what makes you think the rows you've now defined are included in the original construction? you say they must be without offering any proof, other than asserting something that is unjustified (because of you misconception about the axiom of infinity) proof, and i prove they aren't.

it doesn't matter if you even permute rows because i can just unpermute them back to the form you have in the article.

it looks like you've just put 1s in every place above the diagonal, and thus you've exactly got the elements of the power set whose complement contains only finitely many elements, these are called cofinite and are also countable.


so, go through the water tight proof i've offered you twice now in the last 5 posts or so alone (and many times before that) point be point and say where you think it is wrong, and why, and give a counter example based upon the construction you offer in the article newdiagonl.pdf which it has to be said is now worse because you've taken out any mention of how you actually *might* construct the infinite array.

[Organic]

Matt,

What is the result of 2^aleph0 - aleph0?

[matt grime]

if by that what you mean what is the cardinality of a set of cardinality 2^aleph-0 after removing aleph-0 elements, then the answer is that the cardinality of this set is strictly greater then aleph-0. (it is 2^aleph-0 as we will show)

simple example: there is a bijection between between R and R\Z

defein the map piecewise

on [0,1)

send 0 to 1/2, 1/2 to 1/3, 1/3 to 1/4 etc call this map f and extend to the rest of [0,1) by setting it to be the identity

define the map analogously on each interval [n,n-1)


alternatively, for R\N, say, it is still infinite, let x(n) be any sequence in R\N, send

n to x(2n) and x(2n) to x(2n+1)

that do you?

[Organic]

Thank you Matt,

So if you look now at this:

3 2 1 0 3 2 1 0
2 2 2 2 2 2 2 2
^ ^ ^ ^ ^ ^ ^ ^
| | | | | | | |
v v v v v v v v
...0 0 0 0 <--> 1 or ...1 1 1 1 <--> 1
...0 0 0 1 <--> 2 or ...1 1 1 0 <--> 2
...0 0 1 0 <--> 3 or ...1 1 0 1 <--> 3
...0 0 1 1 <--> 4 or ...1 1 0 0 <--> 4
...0 1 0 0 <--> 5 or ...1 0 1 1 <--> 5
...0 1 0 1 <--> 6 or ...1 0 1 0 <--> 6
...0 1 1 0 <--> 7 or ...1 0 0 1 <--> 7
...0 1 1 1 <--> 8 or ...1 0 0 0 <--> 8
...1 0 0 0 <--> 9 or ...0 1 1 1 <--> 9
...1 0 0 1 <--> 10 or ...0 1 1 0 <--> 10
...1 0 1 0 <--> 11 or ...0 1 0 1 <--> 11
...1 0 1 1 <--> 12 or ...0 1 0 0 <--> 12
...1 1 0 0 <--> 13 or ...0 0 1 1 <--> 13
...1 1 0 1 <--> 14 or ...0 0 1 0 <--> 14
...1 1 1 0 <--> 15 or ...0 0 0 1 <--> 15
...1 1 1 1 <--> 16 or ...0 0 0 0 <--> 16
... or ...

or a mixing of them

has a length of 2^aleph0 unique 01 sequences.

[matt grime]

and what? why won't you even attempt to refute the proof offered that you're wrong? if you're so sure of you position it should be quite easy for you.


i'm not sure where you're going but the two lists you've written enumerate the finte and cofinite elements of the power set. there are more sets in the power set than that. in fact it is a trivial exercise to show that the countable union of countable sets is countable, and you're not going to get anywhere with this idea.

come on organic, you think i don't understand infinity, and you're the expert. what's wrong with the proof offered to you?

it is perfectly mathematically sound. you're not the first person to have made this mistake and you won't be the last.

[Hurkyl]

Hurkyl,

This is exactly the beautiful thing in {_} content, for example:

.__. = Finite line = [__]

__ = Infinite line = (__)

.__ = Infinite line = [__)

|{.__.}| = 1

|{.____.}| = 1

|{.________.}| = 1

|{.__}| = 1

|{__}| = 1

Shortly speaking, __ is the essence of an invareant self similarity over scales.

So how do you reconsile this with the fact that, if I assume the real numbers are countable, I can prove the real numbers have a length less than 1/2?



Anyways, one of your major problems is that you seem to confuse the order of the quantifiers in the statements we make.


For example, consider these two statements:

For any (non-bald) person P you pick, I can choose a color C, such that person P's hair is color C.

I can choose a color C, such that for any (non-bald) person P you pick, that person P's hair is color C.


One of these claims is very easy, and one of these is impossible! I hope this demonstrates why the order of these operations is important.


Sometimes, doing things procedurally helps understanding. You fulfill each quantifier one step at a time before moving onto the next one.

For instance, I claim this is possible:

Step 1: You choose any person.
Step 2: I choose a single color.
Fact: The person you chose has hair with the color I chose.

Proof: At step 2, I can look at the person you chose, and select his hair color.

I claim this is not always possible.

Step 1: I choose a single color.
Step 2: You choose any person
Fact: The person you chose has hair with the color I chose.

Proof: At step 2, you know what color I chose, and you can choose a person with a different hair color.


The problem at hand (enumerating the list of binary sequences)

There exists a list L such that for any binary sequence S, S is on the list L.

So you have to do this in steps:

Step 1: You have to choose a list.
Step 2: I choose a binary sequence.
Query: Is the sequence I chose on the list you choose?

Now, the kicker is that step 2 doesn't happen until you've completely specified your list L. Once I've chosen a binary sequence, you can't go back and change your list.

Let me say this again.
You have to specify everything relevant about your list before we start choosing binary sequences.

Let me give an example.

Suppose you give us a specification for a list.
We mention a binary sequence.
You come back with a new specification for a list.

In this example, you have failed. The list must be completely specified before we start choosing real numbers.



And, incidentally, for your latest attempt, I choose the binary sequence:
...010101

that is, the sequence \{x_n\} where x_i = (1 + (-1)^i)/2 (i \in \mathbb{N}), or equivalently, x_i = 1 iff x_i is even.

[Organic]

Matt,

You Wrote:

I've just shown you every row has a finite number of 1's in it.

If this is your proof then it does not hold on this:

3 2 1 0
2 2 2 2
^ ^ ^ ^
| | | |
v v v v
...1 1 1 1 <--> 1
...1 1 1 0 <--> 2
...1 1 0 1 <--> 3
...1 1 0 0 <--> 4
...1 0 1 1 <--> 5
...1 0 1 0 <--> 6
...1 0 0 1 <--> 7
...1 0 0 0 <--> 8
...0 1 1 1 <--> 9
...0 1 1 0 <--> 10
...0 1 0 1 <--> 11
...0 1 0 0 <--> 12
...0 0 1 1 <--> 13
...0 0 1 0 <--> 14
...0 0 0 1 <--> 15
...0 0 0 0 <--> 16
...

because now the next notation in ... is always 1.

You conceptual mistake is this:

You clime that each column is constructed from finite number of 0 or 1 notations, therefore the list is a collection of infinitely many finite structures and therefore it is a countable list.

Also you clime that each length of 0 or 1 sequences along each column must be infinitely long, which means that if we start from 0 notation we will never get 1 notation or if we start in 1 notation (in the above example) we will never get 0 notation, therefore we can conclude that the length of each column must be countable and finite.

My answers are:

1) by using this trick 0 0' 1 1' 2 2' 3 3' 4 4' ( please see http://home.ican.net/~arandall/abelard/math12/Cantor.html )
we can build this list:

3 2 1 0
2 2 2 2
^ ^ ^ ^
| | | |
v v v v
...1 1 1 1 <--> 1 (1)
...0 0 0 0 <--> 2 (1')
...1 1 1 0 <--> 3 (2)
...0 0 0 1 <--> 4 (2')
...1 1 0 1 <--> 5 (3)
...0 0 1 0 <--> 6 (3')
...1 1 0 0 <--> 7 (4)
...0 0 1 1 <--> 8 (4')
...

which is a mixed list of positive-negative sequences of 2^aleph0 length.

2) aleph0 as some "transfinite" object beyond n in N CANNOT EXIST because when we go beyond n in N we are in no information state for any mathematical research (in this case no base value can exists therefore no_base_value^aleph0 is meaningless), as clearly can be shown in this model:
http://www.geocities.com/complementarytheory/RiemannsLimits.pdf

It goes like this:

No base_value --> no notations --> on information --> no Math (--> no "transfinite" universes).

Shortly speaking, the "transfinite" universes do not hold water.

[matt grime]

but you're cheating by writing that, aren't you? that isn't the array you had in the article that you've been using. that array, if it is defined in the obvious way lists the cofinite sets and I can prove by the same argument but switching a couple of details that not all the sets in the power set are there, indeed only a countable number of them are.

I proved that none of the elements on the new list youve written down is on the old one.

You haven't properly specified how to generate that list.


Read Hurkyl's last post about how you fix the selection of elements in the power set (by the construction in the paper where the 1st column is 010101... the second is 00110011..) and THEN we find something not on the list (any infinite element of the power set).


Now you introduce a new list with no finite element of the power set on it.

not every element of the power set is on the two lists; it would take an uncountable set of lists like this to do so.

why do you think you can just create a new list like this?

go back, quote my proof, and after each step say whether you agree or disagree with the deduction and why, offering counter example or counter proof if you think you can.

my proof had nothing to do with the new list you've produced, but it didnt' have to deal with it.


it is only your belief that you can continue to enumerate the rows AND have them be the power set because you think the finite case implies the infinite. that is not true, as this argument proves, and as you yourself prove, but then disregard becuase it doesn't suit your argument. you cannot prove what you want because it isn't true, and your only argument against this is 'well i think it ought to be'.

[matt grime]

just seen your edited post. it contains many more errors


at no point to i state that if the column starts with a 0 it stays 0, in fact i state it is an alternating block of r 0s, then r 1s then r0s and so on, for the rth column.

read the proof, defined for the array as you wrote it in the original form.


your new array interleaves finite and cofinites sets. the set of even integers is not on the list.

[Organic]

Matt,

I proved that the "tranfinite" (what I call an actual infinity) is beyond Math language.

[matt grime]

you haven't rebutted the proof i gave.

all the assertions you make about my 'claims' are wrong seeing as

1. you are the person defining the array

2. you state they [the rows] are countable AND they are all the elements of the power set.

3. i do not claim that any block of 0s or 1s in any column is infinite


3. the r'th column is infinite the entries are defined in alternating finite blocks of 0s and 1s, do you not remember how you defined them? you have expunged the construction from the latest version, whioch means that we actually no longer know what the definition for the infinite array is explicitly.


4. why is it important that my proof doesn't hold on something that it wasnt even defined for?


5. aleph-0 isn't a number, why do you keep pretending that it is? there is not an aleph-0'th place on the list. aleph-0 is the 'cardinality of the set N, it is a definition, that is all.


I do not need to prove anything about this second list as you've defined the entries in the t'th row to all be 1 after the t'th place, hence every row corresponds to a set whose complement is finite.



so, take the proof offered to you repeatedly wuote it and after each line state what you consider to be wrong with it.

I can simply state what's wrong with your proofs - they contain unfounded suppositions about infinite sets.

[matt grime]

tell you what, let's end any doubt

pick one of your diagrams - either the original or the new interleaved one

the list of rows is countable in your opnion, tell me what the row labelled t corresponds to, how does one generate its elements?

[Organic]

5. aleph-0 isn't a number, why do you keep pretending that it is? there is not an aleph-0'th place on the list. aleph-0 is the 'cardinality of the set N, it is a definition, that is all.

Aleph0 as "transfinite" object is beyond Math language.

Therefore any use of in by Math is nonsence.

Cantor started this nonsence, you continue "using" it.

[Organic]

My list cannot be but an ordered list of unique combinations of 01 notations in each row and in each column, where each row is aleph0 width and each column is 2^aleph0 length.

It is simple and clear but the "transfinite" nonsence does not give you the chance to see it.

Aleph0 cannot be but a potential infinity.

[matt grime]

give the method of construction of your infinite list.

how do you enumerate the t'th row?

it looks as though

you send the string (x_0,x_1,x_2.....

with the ordering going backwards from the article's as you read right to left there

to the expansion (\sum x_i2^i) -1

is that what you're doing?

[matt grime]

it should be noted that i'm not saying the power set of N doesn't have 2^alpeh-0 elements, but i have disproved your attempted enumerable construction of it.

[Organic]

Hurkyl,


until you've completely specified your list L.

This is what so beautiful in fractals, the quantity does not matter but the invariant self similarity over scales.

Any form of a^b is a fractal.

My list has a^b form, therefore it is satisfied by its own self similarity (again quantity does not matter).

And, incidentally, for your latest attempt, I choose the binary sequence:
...010101

Contors diagonal is an aleph0 width of my list, therefore ...010101 is already in the list, which its length is 2^aleph0.

[Organic]

My list has a width of arithmatic row of (if base 2) 0,1 notations,
and a length of geometric column of 0,1 notations.

Width and length are non-finite.


but i have disproved your attempted enumerable construction of it.

You disproved nothing.

[matt grime]

so let's suppose 01010101... (the even numbers in N) is on the enumerated portion of the list.


it must be at some point r for some r in N, say: the r'th row.


r cannot be one of the rows you've now put in with all 1's eventually, so it must be one of the other ones. look how they're constructed.. oh, see that every entry is eventually zero so 010101... can't be on of the enumerated rows.


so we can just consider the diagram without the infinite row of ones, and go back to the first diagram you have.

ever column after the rth has at least r zeros at the beginning and therefore cannot contribute a 1 to the r'th row in this diagram.

brick and wall spring to mind


do you even accept that given a string of 0s and 1s and that if the list is enumerable we can just consider the row it corresponds to?

[Hurkyl]

How do you deal with the fact that if the real numbers are assumed to be a countable set, that I can prove their total length is less than 1/2?


Contors diagonal is an aleph0 width of my list, therefore ...010101 is already in the list, which its length is 2^aleph0.

What is the number of the row which contains it?

(Incidentally, due to the way you wrote it, I don't even know what is the number of the row that contains the all ones sequence!)

[matt grime]

Originally posted by Organic
My list has a width of arithmatic row of (if base 2) 0,1 notations,
and a length of geometric column of 0,1 notations.

Width and length are non-finite.


You disproved nothing.

show me where the proof goes wrong? quote it word for word and point out exactly where the proof is wrong for the list as defined in your article.

Better yet in this forum explicitly define how the list is formed?


I cannot offer to show where your proof that the list is both countable and contains all the elements of the power set goes wrong because you do not prove that it eumerates all of the power set, just state it must.


Come on, quote that proof, and explain the flaw you think there is in it. simple, couldn't be simpler?

[Organic]

Hurkyl,

What is the number of the row which contains it?

First Please give me the number (the index if you like) of each prime number.


Please explain me again without professional notations the argument about 1/2.

Thank you.

[Organic]

,

Come on, quote that proof, and explain the flaw you think there is in it. simple, couldn't be simpler?

Please understand this, the invariant self-similarity of a fractal does not depend on quantity, which means that any part of it (its local level) is equal to "all" of it (the global level of it).

So in a fractal finite and infinite are satisfied by the invariant structural property, and quantitative property is not important.

This is the deep meaning of cardinality in a fractal, not its size but its invariant self similarity.

If you understand this then you have a gate to my world.

[suyver]

This thread is sooooo funny! [:D]

If I didn't know better, I'd say that Matt and Hurkyl take turns pretending to be Organic and posting nonsence just for fun.

[Organic]

suyver,

You are invited to add your joke (please say some funny thing on fractal-like nanocrystals).

[matt grime]

and those counter claims state what? apart from that you don't seem to able to either understand the concept of proof or a simple request to demonstrate where you think the proof offered to you goes wrong?


here is the proof based upon the array as written in your article


the first column is on the right, then going to the left we number the columns accordingly.

the first column is 'based on 2^1' and goes 0101010101....

the second colummn reads, downwards, 00110011....

and so on for each column.

the r^th column starts with r zeroes.
then r ones then r zeroes...
Yes?

Let t be any row, the entry in the s'th place (reading right to left) is the entry from the s'th column.

whenever s >t this entry must be zero.

therefore any row is eventually all zeroes, and every element you enumerate in the power set is finite.

Read that argument carefully and don't dismiss it simply because it is me. as you keep telling people they must be open to new ideas, well, so must you.


edit: also recall the worked examples i gave for the case of the first to 5th rows

so go through it and at each point tell me what you think is wrong with the deduction there.



you've now interleaved the confinite sets, but it's trivial to show that they are countable and contain only sets whose complement is finite. these two observations combined demonstrate your enumerated list of elements of the power sets contains none of the uncountable number of subsets of N which are infinite and whose complement is infinite.


so quote all this and go through this step by step and write at each stage whether you agree or disagree, and if you disagree, why.


you do not need to cite fractals or prabability, these have no bearing on this result. And just saying 'but they do, and you're wrong' only demonstrates your frailty of position.


and i wish you were right suyver - i don't know why i spend so much intellectual energy on this topic.

a goood thing to note is that organic never posts in other people's threads unless it is to tell them about how his theory applies there. surely if he cared and knew mathematics or physics he'd want to help other people too and answer their queries about maths?

[suyver]

Originally posted by Organic
suyver,

You are invited to add your joke (please say some funny thing on fractal-like nanocrystals).

pass.

[Organic]

Matt,

This is useless, you are one of the prisoners in "Plato's Allegory of the Cave" http://faculty.washington.edu/smcohen/320/cave.htm .

And the name of your jailer is Cantor.

[matt grime]

Originally posted by Organic
Matt,

This is useless, you are one of the prisoners in "Plato's Allegory of the Cave" http://faculty.washington.edu/smcohen/320/cave.htm .

And the name of your jailer is Cantor.

if the strength of your position is so evident, and you are this genius that you seem to purport to be why can you not respond to the simple request to go through the proof offered step by step and explain your thoughts about it.


failing that present an explicit construction of the array you imply exists, and demonstrate that every element of the power set is enumerated.

define the bijection between the rows and N. I have offered you an example and you've ignored it.

explain why it is that you make these claims about the case for N based upon the cases for the finite sets without citing the axiom of infinity induction.

you are attempting to prove that mathematics is inconsistent, and thus you must do so from within mathematics, that is why i do not need to consider your assertions about fractals and probability. they are irrelevant to the discussion in hand as you've not proved that there is a problem within mathematical treatments of N that don't deal with them. moreoever, you have not defined fractals or probabilty without using the mathematics that you consider to be flawed.

[Organic]

Matt,

My allegory to you:

Matt: “Define a cat”.

Organic: Taking a cat and put it in front of Matt, then says “here is a cat”.

Matt: “No, define a cat”.

Organic: “Matt, it is in front of you”.

Matt: “You don’t understand, define a cat”.

Organic: ”what is define?”.

Matt: “Take a knife cut the cat to pieces and define each piece by putting it back to its place”.

Organic: ”But then you have no alive cat but pieces of flash. For me a cat is first of all alive thing in a one organic piece”.

Matt: “life is not important, definition is important, define a cat”.

[matt grime]

i didn't think you liked allegories.


ok, put in front of me the array you claim has countably many rows that enumerate the power set of N.

Explain the rule for generating it. You say it must have a certain property by construction, what is the construction?

explain how you are bijecting to N, explain why then you manage to ignore the fact that that implies it does not enumerate the power set. remember you are only allowed to use my mathematics because that is what you are claiming is incomplete and cannot handle infinity.

Putting a cat in front of me doesn't define a cat - it gives me an example of a cat. If it were an abyssinian, would I then have to only accept that abyssinians were cats, and that, say, a siamese weren't a cat because it doesn't look like an abysinian?


Now take the proof of mine and carefully go through it and explain where it is wrong.

i've done that for you - you're argument is wrong because you're basing *assumptions* about it on the finite case, that are meaningless in the infinite.

[Organic]

Matt,

You don't know what is infinity, no more no less.


When you know it then and only then we shall communicate.


Bye.

[matt grime]

so you can't find a flaw in the proof then?

or can't you explain how to generate this infinite array?

which of all the observations on the inadequacy of your mathematics is causing you the most concern?

or all of the above?

[suyver]

And the jury declares Matt Grime the winner!
The crowd goes wild!

[:D]

[matt grime]

I will concede this - I cannot state 'infinity is ....' and fill in something for the ... that is a definition in anyway that is very satisfactory. No mathematician would, or could without qualifying their statement. There was an interesting thread on sci.math about the role of infinity in mathematics, and the consensus was that mathematicians whilst using the term to illustrate concepts, would, when pressed to be rigorous, switch to another definition.

For instance, when we say there is an 'infinity of' possibilities, we actually mean, there is not a finite number of possiblities; given any finite number of options I can find another one'. When we say x(n) tends to x as n tends to infinity, what we actually mean is a statement that at no point includes the word infinity. Then there's the case of the sum to infinity, which is just the limit of a sequence as above, again with no infinity mentioned. Then the sum is 'infinite' if it is not finite, if there is no limit in the sequence of finite sums, that's all, agian we don't actually have an infinity there do we? Of course there is the point at infinity of the riemann sphere which neatly encapsulates the idea of being 'not finite', and which allows us to do many useful analytic operations. It is often called infinity, and can be related to the other examples, but is it 'infinity'? No, just like things such as multiplication it is contextual - the multiplication of real numbers isn't the multiplication of matrices is it? In short infinity is a useful concept, just as continuity is, but there is no object one can satisfactorily point to as infinity, just as there is no object one can point to and say that object is continuity.

Many cranks have this idea that infinity is actually something, something tangible, and that when we say the sum from one to infinity, we actually mean sum all the finite bits and then stop AT infinity just like we can stop at 7 or 20,445. If people learned the distinctions about these things we'd all be a lot better off. All this is compounded by the teaching that 1/0 IS infinity. It isn't, it is undefined in the ordinary arithmetic that they know, but it is true that 1/x can be made arbitrarily large, which is not the same thing at all.

[Deeviant]

Originally posted by Organic
Matt,

My allegory to you:

Matt: “Define a cat”.

Organic: Taking a cat and put it in front of Matt, then says “here is a cat”.

Matt: “No, define a cat”.

Organic: “Matt, it is in front of you”.

Matt: “You don’t understand, define a cat”.

Organic: ”what is define?”.

Matt: “Take a knife cut the cat to pieces and define each piece by putting it back to its place”.

Organic: ”But then you have no alive cat but pieces of flash. For me a cat is first of all alive thing in a one organic piece”.

Matt: “life is not important, definition is important, define a cat”.

A interesting analogy. Like the cat, your position would not survive a thorough disection, unlike the cat, your position wasn't a living breathing thing cute furry thing before the process.

[Hurkyl]

Let us assume that there exists a list of all real numbers.

I wish to create a list of intervals such that every real number is in one of these intervals.


For my first interval, I will look at the first real number in the list, and then choose an interval of length 1/4 that contains that real number.

For my second interval, I will look at the second real number in the list, and then choose an interval of length 1/8 that contains that real number.

etc...

For my n-th interval, I will look at the n-th real number in the list, and then choose an interval of length (1/2)^(n+1) that contains that real number.


So now I have a list of intervals.


Now, every real number is in one of these intervals.
Proof: Pick any real number. It appears somewhere in the list of all real numbers we selected at the beginning; let's say it appears in the n-th position. Well, the n-th interval was created so it contains the n-th real number in the list, therefore the real number we picked is in one of my intervals.


(Aside: Notice how I gave a set of instructions on how to create my list of intervals before I started proving things about my list. And notice that I'm not going back and altering those instructions or adding new instructions now that I have started proving things)


So this collection of intervals covers the entire real line.


Now, let's look at the length of the intervals:
The first one has length 1/4.
The second one has length 1/8.
The third one has length 1/16.
etc.
The n-th one has length (1/2)^(n+1) for all n.

The sum of all of these lengths is 1/2.


Now, one of the nifty things about length (and similarly about area and volume ) is that if you have a list of "shapes", then the length covered by those shapes cannot be bigger than the sum of the lengths of those shapes!

So, this means that the length of the real line cannot be bigger than 1/2!

[Organic]

Hi Deeviant,



unlike the cat, your position wasn't a living breathing thing cute furry thing before the process.

Why not?

[Organic]

Dear Hurkyl,

I hope that by this post we (maybe for the first time) will communicate between our different perceptions about the infinity concept.

First let us take the model of a line and I mean a smooth line without any points or segments (what you call intervals) included in it.

Zoom-in 1x2, 1x3, 1x4, 1xn, 1xn+1 ... and you find the invariant self similarity 1.

In a more formal way |{__}| = 1.

Now, let us talk on what you call the "real-line" of R collection.

The real line of R collection is not {__} form but {...} form.

Shortly speaking, we can find unique elements only in {...} form.

Only in {...} form we can find a one-to-many relation , and if we take your private case of 1/4, 1/8, 1/16, ... then our one-to-many relation has the invariant 1/2 which is exactly the invariant of a Binary Tree on infinitely many scales of it.

Shortly speaking, R collection cannot be but a {...} form.

Your conceptual mistake is that you take R collection as {__} form.

But {__} is a representation of what I call an "actual infinity", and cannot be used as an available information for Math language, or in other words, it is the strong limit of Math language.

Shortly speaking, R collection cannot use the word "line" because no infinitely many points or segments(=intervals) can be a solid line.

From a symmetry point of view, the opposite of the strong limit {__} is the weak limit, which is notated as {} content.

Please look again my major theorem: http://www.geocities.com/complementarytheory/Theory.pdf

And please give your detailed remarks.

Yours,

Organic

[Hurkyl]

First off, length is not preserved by scaling; a scaling factor of 2 means that lengths are doubled, areas are quadrupled, volumes are octupled, et cetera.

In particular, the line segments [1, 2] and [1, 4] may be similar, but there's a scaling factor of 3, so corresponding figures do not have the same lengths.


My currest best guess as to what you're trying to say is that you're either trying to make a distinction between a geometric line and the set of points incident with that line, or you're trying to conceptualize the distinction between the ideas of the "set of real numbers" and the "topological space of real numbers."

I have to go to work so I don't have time to expound upon this at the moment.

[matt grime]

why do you, organic, not accept the simple assertion that there is not bijection between the Natural numbers and R? Why do none of the proofs offered satisfy you? Why? It's not difficult; these things are just defininitions, and not very hard ones at that.

[Organic]

I agree that |R|>|N| but both are countable because R collection cannot be a solid line.

[Hurkyl]

The word "countable" is defined as:

A set S is countable if and only if there exists a function from the natural numbers onto S.

The word "uncountable" is defined as:

A set S is uncountable if and only if it is not countable.

The relation > is defined as:

|S| > |T| if and only if there does not exist a function from T onto S.


So why do you accept |R| > |N| but not R is uncountable.

[Organic]

Hurkyl,

I am talking about the invariant self similarity of the structural property of {__} and/or {...} where quantity is not important at this stage.

First off, length is not preserved by scaling; a scaling factor of 2 means that lengths are doubled, areas are quadrupled, volumes are octupled, et cetera.

Length measurement depends on comparison between some constant value and some changeable value (I am talking about 1 obserber and one object at a time).

If the observer is the constant then we say: "the observed object is changed".

If the object is the constant then the observer is changed.

Let us say that in our case the observer is changed.

In this case observer x 1/2 --> object x 2 , observer x 1/3 --> object x 3, ... , observer x 1/(n+1) --> object x (n+1)

So, in this case the observed object length = 1.

If there are at least two different objects, the length changes between them are not depend on the observer but on one of the objects, which is used as the constant value 1.

But again the first thing here is not the length changes but the invariant structural self similarity of {___} and/or {...} forms.

My currest best guess as to what you're trying to say is that you're either trying to make a distinction between a geometric line and the set of points incident with that line, or you're trying to conceptualize the distinction between the ideas of the "set of real numbers" and the "topological space of real numbers."


My point of view disagree with Contor's point of view about a collection with cardinality aleph-1 of infinitely many objects (={...} form) that can construct a solid line (={___} form), as we can find here: http://mathworld.wolfram.com/LineSegment.html

Distinction is only the first step.

The second step is to combine between {__} and {...} forms and the result is:

http://www.geocities.com/complementarytheory/ASPIRATING.pdf


So why do you accept |R| > |N| but not R is uncountable.

http://www.geocities.com/complementarytheory/NewDiagonalView.pdf

Definitions are only tools of meaning, if they contradict the meaning then we have to find another definitions that can express the meaning.

In this case "countable" and the "must have" connection with N is a conceptual mistake.

Shortly speaking:
"A set S is countable if and only if there exists a function from the natural numbers onto S."

Is a conceptual mistake.

[matt grime]

Right, you say we must be open to your ideas, but you won't listen to us.
Definitions must not contradict what we know, yet in your newdiagonalpdf you define an array using some (undefined) construction that I've shown doesn't have the properties you define it to have.

Listen to your own advice.

So, where does Hurkyl's measure theoretic proof go wrong for showing there is no bijection from N to R? And don't start this invariant self similarity crap again because that isn't what the proof is using. We can dress it up some more if you don't understand measure theory, and use Baire's Category theorem - a one point set is nowhere dense, hence the countable union of them is nowhere dense and thus cannot be R.

[Hurkyl]

The thing is, sets do not have any structural quantities.

The only thing that matters about a set is the elements it contains.

And up to bijection1, the only property of a set is its cardinality2.



So, basically, when you're using set theory to describe things, at the ground floor, quantity is the only thing that matters3. Once you have a set, you then impose structure on that set. For instance, you can impose an order on a set, and the result is called an ordered set So, for example:

(R, <) is an ordered set4
(where R stands for the set of real numbers, and < stands for the usual ordering on them)

Now, if we're being entirely proper, we should say "(R, <) is an ordered set" or "< is an ordering for R", not "R is an ordered set" or "R is an ordered set with ordering <" The reason is that the structure of order is something that is beyond the meaning of the term 'set'. That being said, mathematicians often abbreviate themselves by using "R" when they should be using "(R, <)" for economy of notation.



Another structure we can impose on a set is that of being a field. For example

(R, +, *) is a field4.
(Where R stands for the set of real numbers, and + and * stand for the usual addition and multiplication operation on them)

Again, a mathematician will often abbreviate himself by saying "The field R" instead of saying "The field (R, +, *)". Anyways, the important thing is that the structure of + and * are not part of the set of real numbers; they are additional structure imposed beyond that of being a set.


For one more example, there is the ordered field (R, +, *, <). This is yet another structure that is different from the previous ones. The ordered set (R, <) does not have arithmetic, and the field (R, +, *) is not ordered.


To say it again, the point I'm making is that sets do not have structure! All sets have are elements! If you are using a structure on some set, that is something that is beyond its definition as a set!

To state my point in a different way, if you don't the notion of "quantity" and "elements" to be there from the very beginning, you can't use a set theoretic description of your ideas.

Hurkyl



1: In laymen terms, this means that if we consider two sets to "the same" if there is a bijection between them.

2: and any property that can be derived knowing nothing about the set but its cardinality, such as the property of being finite (cardinality equal to a natural number) or infinite (cardinality not equal to a natural number)

3: More precisely, it is cardinality that matters.

4: I am being somewhat narrow here for clarity. This is the set theoretic way to write a field; the theory of fields does not have things like ordered triples or sets of elements! Also, even when using ZF, there exist fields for which a set of all elements does not exist, such as the surreal numbers. Also, instead of what I wrote, one would often write (R, +, *, 0, 1), putting the additive and multiplicative identities into the notation.

[Hurkyl]

My point of view disagree with Contor's point of view about a collection with cardinality aleph-1 of infinitely many objects (={...} form) that can construct a solid line (={___} form)

The idea that lines have points on them goes back at least to Euclid; I imagine much further. [:)]

I imagine the concept that lines are made out of points is just as old, but modern mathematics dosen't require this point of view. In fact, depending on the circumstance, mathematicians can be downright pedantic about insisting that you keep the ideas of "a line" and "the points on a line" seperate.



Definitions are only tools of meaning, if they contradict the meaning then we have to find another definitions that can express the meaning.

In this case "countable" and the "must have" connection with N is a conceptual mistake.

Shortly speaking:
"A set S is countable if and only if there exists a function from the natural numbers onto S."

Is a conceptual mistake.


Mathematics need a phrase that means "There is a function from N onto S". The word countable was chosen. Thus, when a mathematician means "there is a function from N onto S", we use the phrase "S is countable", and when a mathematician uses the phrase "S is countable", we mean "there is a function from N onto S".

Mathematicians are using the definition of the phrase "S is countable" as a tool to express the meaning "There is a function from N onto S".


So, I'm not sure what your objection is. Do you just not like the word mathematicians have chosen for this purpose? If we used the term "S is gazorninplat" to mean "there is a function from N onto S", would it be acceptable to you? If so, then R is not gazorninlpat, and anytime you see a mathematician use the word "countable" you should mentally substitute the word "gazorninplat", and be happy!

[matt grime]

R is not countable. Please prove it is. None of the things you've claimed so far is a proof.

You say complementary multiplication is not commutative. I would question the assertion that it is a multiplcation. Not because it is not commutative, but because it is not a binary operation from NxN to N. It is an operation on certain combinatorial structures. I might call them trees but you don't believe in the usual definition of tree. For instance, you calculate 2*3. What is the outcome? Which number is it? Moreover, as you've challenged us to produce complementary additive and multiplicative operations (on N) would you mind stating what complementary means in this sense, and why we must define it on N when you don't define it on N.

And if you dismiss Hurkyl's use of the word structure as unimportant, how come you use the same word in your quote from one of your writings?

You don't like N in countable? Why not? the idea of bijection with N and the reason you have the word 'count' in there is because then you can label the elements with the counting numbers.

If you don't like that then we can say a set is countable if it is bijection with:

1. Q

2. Z

3. The set of primes

4. The set of even integers

5. The set of polynomials with coefficients in F_2.

6. The set of spheres in R^3 whose centres lie at rational coordinates and whose radius is an integer.

7. The set of all finite groups. (Up to isomorphism. Or the set of all finite sets up to isomorphism in the category SET)



Anyway, here's another question you've still not answered.

Why does the fact that the arrays with columns labellled 1 to n and rows labelled 1 to 2^n forms a complete description of the power set of 1,..,n imply that the case for an infinite countable set also has a countable number of rows?


You keep saying 'by construction'. What construction? I don't see one in your latest version. You had one once, and using that we've proved again and again that you've made a mistake. Now you don't even attempt to describe the construction. Why not? What is it?

[Organic]

Hurkyl,


The thing is, sets do not have any structural quantities.

There is no such a thing structuarl quantitiy, and I never wrote such a thing.

I wrote structoral property, and a set has structural property, please take for example the Von Neumann Hierarchy:

0 = |{ }| (notation = {})

1 = |{{ }}| (notation = {0})

2 = |{{ },{{ }}}| (notation = {0,1})

3 = |{{ },{{ }},{{ },{{ }}}}| (notation = {0,1,2})

4 = |{{ },{{ }},{{ },{{ }}},{{ },{{ }},{{ },{{ }}}}}| (notation = {0,1,2,3})

and so on.

A set is only a framework to explore our ideas.

The concept of oreder set does not depend on the quantity concept as can clearly shown here:

By Complementary Logic multiplication is noncommutative,
but another interesting result is the fact that multiplication
and addition are complementary opreations that can be ordered
by different symmetry degrees where quantity remains unchanged
for example:

A Number is anything that exists in ({},{__})

Or in more formal definition:

({},{_}):={x|{} <-- x(={.}) AND x(={._.})--> {_}}

Where -->(or <--) is ASPIRATING(= approaching, but cannot become closer to).

If x=4 then number 4 example is:

Number 4 is a fading transition between multiplication 1*4 and
addition ((((+1)+1)+1)+1) ,and vice versa.

This fading can be represented as:


(1*4) ={1,1,1,1} <------------- Maximum symmetry-degree,
((1*2)+1*2) ={{1,1},1,1} Minimum information's
(((+1)+1)+1*2) ={{{1},1},1,1} clarity-degree
((1*2)+(1*2)) ={{1,1},{1,1}} (no uniqueness)
(((+1)+1)+(1*2)) ={{{1},1},{1,1}}
(((+1)+1)+((+1)+1))={{{1},1},{{1},1}}
((1*3)+1) ={{1,1,1},1}
(((1*2)+1)+1) ={{{1,1},1},1}
((((+1)+1)+1)+1) ={{{{1},1},1},1} <------ Minimum symmetry-degree,
Maximum information's
clarity-degree
(uniqueness)


============>>>

Uncertainty
<-Redundancy->^
3 3 3 3 | 3 3 3 3
2 2 2 2 | 2 2 2 2
1 1 1 1 | 1 1 1 1 1 1 1 1 1 1
{0, 0, 0, 0} V {0, 0, 0, 0} {0, 1, 0, 0} {0, 0, 0, 0}
. . . . . . . . . . . . . . . .
| | | | | | | | | | | | | | | |
| | | | |__|_ | | |__| | | |__|_ |__|_
| | | | | | | | | | | |
| | | | | | | | | | | |
| | | | | | | | | | | |
|__|__|__|_ |_____|__|_ |_____|__|_ |_____|____
| | | |
(1*4) ((1*2)+1*2) (((+1)+1)+1*2) ((1*2)+(1*2))

4 = 2 2 2
1 1 1 1 1 1 1
{0, 1, 0, 0} {0, 1, 0, 1} {0, 0, 0, 3} {0, 0, 2, 3}
. . . . . . . . . . . . . . . .
| | | | | | | | | | | | | | | |
|__| |__|_ |__| |__| | | | | |__|_ | |
| | | | | | | | | | |
| | | | |__|__|_ | |_____| |
| | | | | | | |
|_____|____ |_____|____ |________| |________|
| | | |
(((+1)+1)+(1*2)) (((+1)+1)+((+1)+1)) ((1*3)+1) (((1*2)+1)+1)

{0, 1, 2, 3}
. . . .
| | | |
|__| | |
| | | <--(Standard Math language uses only this no-redundancy_no-uncertainty_symmetry)
|_____| |
| |
|________|
|
((((+1)+1)+1)+1)


Multiplication can be operated only among objects with structural identity,
where addition can be operated among identical and non-identical
(by structure) objects.

Also multiplication is noncommutative, for example:

2*3 = ( (1,1),(1,1),(1,1) ) , ( ((1),1),((1),1),((1),1) )

3*2 = ( (1,1,1),(1,1,1) ) , ( ((1,1),1),((1,1),1) ) , ( (((1),1),1),(((1),1),1) )

[Hurkyl]

If there is a function from N to S, then there certainly must be a function from N to S...

I don't see what the problem is. Anyways, I'll give a better response later!

[matt grime]

The huge great problem that you have never overcome is that you at no point prove that the rows you enumerate are in bijection with the power set of N.

So.

1. Write down here in this forum the construction you use to generate the r'th row of the array for the ones labelled 1,2,3,, no need to do it for the 1',2' etc


2. Explain where the proof that this enumerates only the finite and cofinite sets is wrong. I've posted it at least 4 times in this thread and you've not said what's wrong with it beyond asserting that the rows mus by construction be the power set, yet you've not defined the construction.



3. Here we go again, let's od it one step at a time.

Just take the diagram as in Newdiognal.

the first column is the alternating sequence 0101010101...
the second goes 001100110011...
the rth goes in alternating blocks of r 0s and r 1s.


Is that correct? and all the rows so produced are in correspondence with the power set of N, aren't they?

Just answer this for now

[matt grime]

Take the array as written in Newdiagonal.

is it true that the construction you allude to is to write the right hand column as 01010101... alternating 0s and 1s, the next column is 00110011.. alternating pairs of 0s and 1s, and that the rth row is r 0s then r 1s then r 0s and so on?

We can ignore the rows yo've now added in as they only correspond to the countable set of cofinite subsets.

[Organic]

Hurkyl,


"There is a function from N onto S". The word countable was chosen.

My problem is not the word "countable" but the "must have" connection between this word and N.

R is also countable and it is defenetly not N because |R|>|N|.

The proof that R is countable can be found here:

http://www.geocities.com/complementarytheory/NewDiagonalView.pdf

If your problem is that I use the notations of N to mark each unique 01 row sequence,
then you can use notations tricks, for example:

By using this trick 0 0' 1 1' 2 2' 3 3' 4 4' ( please see http://home.ican.net/~arandall/abelard/math12/Cantor.html )
we can build this list:

3 2 1 0
2 2 2 2
^ ^ ^ ^
| | | |
v v v v
...1 1 1 1 <--> 1
...1 1 1 0 <--> 1'
...1 1 0 1 <--> 2
...1 1 0 0 <--> 2'
...1 0 1 1 <--> 3
...1 0 1 0 <--> 3'

0 0' 1 1' 2 2' 3 3' 4 4'... is good for base 2.

For base 3 the trick is: 0 0' 0'' 1 1' 1'' 2 2' 2'' 1'' 3 3' 3'' 4 4' 4''...

For base 4 the trick is: 0 0' 0'' 0''' 1 1' 1'' 1''' 2 2' 1'' 3 3' 3'' 3''' 4 4' 4'' 4'''...

And so on ...

Shortly speaking cardinality is first of all based on unique structural differences between elements that have to be notated by equivalence structural periodic repetitions of notations that related to these structural differences.

It means that from structural point of view this mapping

(structural periodic notations = 1)
1 <--> 1
2 <--> 2
4 <--> 3
3 <--> 4
5 <--> 5
8 <--> 6
7 <--> 7
6 <--> 8
...

can be done, where this mapping

(structural periodic notations = 1 <--> structural periodic notations = 2)
1 <--> 1
2 <--> 1'
3 <--> 2
4 <--> 2'
5 <--> 3
6 <--> 3'
7 <--> 4
8 <--> 4'
...

cannot be done.

Shortly speaking, structural property is reacher and much more interesting then quantity property.

I'll say it again, mapping between elements must be checked first of all by the structural propery that exists (or does not exist) between these elements.

By structual point of view aleph0 < 2^aleph0 < 3^aleph0 ... < n^aleph0 < (n+1)^aleph0

By quantitative point of view aleph0 < 2^aleph0 = 3^aleph0 ... = n^aleph0 = (n+1)^aleph0

[matt grime]

Is there any reason why you will not give a direct answer to a direct question?

Is that construction above that I sepcified the construction you are using in Newdiagonal.pdf?

The problem is that each row you've marked with an integer, depending on the version you're now using, contains either finitely many 1s or finitely many zeroes and is therefore not in bijection with the power set.

You've not offered to explain the construction. Until such time as you explain the construction you have nothing concrete to work with. Note you cannot deduce anything by induction on the finite case; that is not how induction works.


Note I wish to amend my observation of the construction to state the rth column has alternating blocks of 2^r 0s and 1s. This doesn't affect the deductions about the construction since 2^r>r

[Hurkyl]

There is no such a thing structuarl quantitiy, and I never wrote such a thing.

It was a typo. I meant to say "structural property".


I wrote structoral property, and a set has structural property, please take for example the Von Neumann Hierarchy:

What about it?


Shortly speaking cardinality is first of all based on unique structural differences between elements that have to be notated by equivalence structural periodic repetitions of notations that related to these structural differences.

No, cardinality is based on the idea of "set" and "invertible function", nothing else. (though it is a generalization of the idea of "quantity" which, again, is not a structural idea)



The concept of oreder set does not depend on the quantity concept as can clearly shown here:

I don't see any ordering given in your example. Anyways, an ordered set is defined to have a set and an order. We can talk about "quantity" in relation to sets, so we can talk about "quantity" in relation to ordered sets.



Shortly seaking, structural property is reacher and much more interesting then quantity property.

Yes, structural properties are generally richer than the "quantity property". However, the point still stands that no matter how much structure you pile upon a set, it is still a set, and thus we can still talk about the "quantity property" of the set.

[matt grime]

Acutally I think you'll find that r^(aleph-0)=s^(alpeh-0) for all r and s in N.

[matt grime]

Originally posted by Organic
Matt,

How my 01 matrix can be finite if there are infinitely many columns and infinitely mant rows?

Where on earth do you think I state the number of rows and columns is finite? I said that any row has either a finite number of 1s in it or a finite number of 0s. That does not say that a row is finite in length

[Organic]

Matt,

How my 01 matrix can be finite if there are infinitely many columns and infinitely many rows where it's width's magnitude=aleph0 and it's length's magnitude=2^aleph0?

[Hurkyl]

By structual point of view aleph0 < 2^aleph0 < 3^aleph0 ... < n^aleph0 < (n+1)^aleph0

By quantitative point of view aleph0 < 2^aleph0 = 3^aleph0 ... = n^aleph0 = (n+1)^aleph0

There is no structural point of view! Cardinalities ignore ALL "structure"!

The only point of view where 2^aleph0 < 3^aleph0 is one where you've actually changed the definition of the symbols!

[matt grime]

Originally posted by Organic
Matt,

How my 01 matrix can be finite if there are infinitely many columns and infinitely many rows where it's width's magnitude=aleph0 and it's length's magnitude=2^aleph0?

Erm, no sorry, at no point have I specified that the matrix is finite. I've told you how to make the matrix up using finite blocks repeated an infinite number of times.

In fact I'm the only one who's specified how to construct it. Why don't you correct that, and tell me what the entries in the r'th row are, and how they are organized.

[Organic]

Matt,

Do you agree that the length of my matrix has a magnitude of 2^aleph0 where its width has the magnitude of aleph0?

You make a big conceptual mistake is you think thet there
must be infinitely many 0 AND 1 notations in each row, because in this case no one of these rows has its unique data.

It means that each row can be reduced to ...1010101010 or ...0101010101 data.

[Hurkyl]

Do you agree that the length of my matrix has a magnitude of 2^aleph0 where its width has the magnitude of aleph0?

It has aleph0 rows, because it is a list.

You make a big conceptual mistake is you think thet there
must be infinitely many 0 AND 1 notations in each row, because in this case no one of these rows has its unique data and can be reduced to infinitely many rows of ...1010101010... data.

So ....011011011 never appears in your list?
What about ...100001000100101?

[matt grime]

Your last post makes little sense - you state that it is possible for the rows to correspond to the power set of N and for there to be no row corresponding to the even integers.

By implication the sets of rows and columns both have cardinality aleph-0, since you enumerate them.

You also claim that the rows are in correspondence with the power set of N, and thus the rows of the matrix must be a set of cardinality 2^alpeh-0.

My contention is that if you enumerate the rows as you do, whereby for row n you take the decomposition of n-1 in binary expansion and write the coeffs going from right to left, that the rows you enumerate do not form a set that is in correspondence with the power set - rather obviously they are exactly in correspondence with the finite subsets. You then try to muddy the waters by including the cofinite sets, without noticing that you are thus contradicting your earlier held beliefs.


I do not state that every row must have infinitely many 0s and 1s. But that there must be some rows (uncountably many as it happens) that do contain infinitely many 0s AND 1s, if this is to be all of the power set. There is none in the enumerated part of your 'matrix'


Now, are you going to explain the construction of the 'matrix', so far you've not offered any way of constrcuting it.

[Organic]

Hurkyl,

Please this time pay attention to the word "each".

You make a big conceptual mistake is you think thet there
must be infinitely many 0 AND 1 notations in each row, because in this case no one of these rows has its unique data.

It means that in ths case (which is not my case, where both forms exist) each row can be reduced to ...1010101010 or ...0101010101 data.

[Organic]

Matt,

I do not state that every row must have infinitely many 0s and 1s. But that there must be some rows (uncountably many as it happens) that do contain infinitely many 0s AND 1s, if this is to be all of the power set. There is none in the enumerated part of your 'matrix'

Both cases must exist in my 01 matrix.

Simply there is no other way.

[matt grime]

Originally posted by Organic
Hurkyl,

Please this time pay attention to the word "each".

You make a big conceptual mistake is you think thet there
must be infinitely many 0 AND 1 notations in each row, because in this case no one of these rows has its unique data.

It means that each row can be reduced to ...1010101010 or ...0101010101 data.


No one is saying (apart from you here) that there must be infinitely many 0s and infinitely many 1s in every row, indeed you don't produce a single row where that is true.

what on earth does it mean to 'reduce' a row to ..0101010 or ..1010101?

no one of these rows has its unique data? eh? That looks like a meaningless sentence.

[Hurkyl]

"each" might be the only word in that sentence that makes sense! (Ok I'm exaggerating, but only slightly)


Does ...011011011 appear in your list?
Does ...100001000100101 appear in your list?
Does ...01010101 appear in your list?
Do you consider these three sequences different?

[matt grime]

Originally posted by Organic
Matt,

Both cases must exist in my 01 matrix.

Simply there is no other way.

Why *must* there be all of the strings with infinitely many 1s and infinitely many 0s in them? Why? Give me one reason. If you are about to say 'by cosntruction' which construction, you've not offered one? SO which construction - just tell me the r'th entry in the t'th column, that's all, even if it's by some inexplicit way like my comment that it appears to be

the r'th column is the sequence of alternating blocks of 2^r 0s then 2^r 1s, you may then read off the rows.

It's a simple request. Note you cannot say by the axiom of infinity induction on the power value' because that is not acceptable; oh you're about to cry foul and say i'm restricting you, but as you've never explained what the axiom of infinity of induction is you can't use it. Induction tells us something is true for an infinite number of cases, it does not tell us what, if anything, is true for an infinite set in the index if that even makes sense. For instance one can easily define the n'th fibonacci number and get an explicit value for it by induction. what would it mean to even talk of the aleph-0'th fibonacci number?

[Organic]

Matt,

In this private case of 01 matrix we can distinguish between to cases of uniquness.

Case 1: there are infinitely many 0 AND 1 notations in each row with a unique order of 01 combinations.

Csae 2: there are finitely many 0 XOR 1 notations in each row with a unique order of 01 combinations.

Only case 1 can be reduced to ..0101010 or ..1010101 where by reduced I mean that we don't care about the order of 01 combinations.

[matt grime]

Originally posted by Organic
Matt,

In this private case of 01 matrix we can distinguish between to cases of uniquness.

Case 1: there are infinitely many 0 AND 1 notations in each row with a unique order of 01 combinations.

Csae 2: there are finitely many 0 OR 1 notations in each row with a unique order of 01 combinations.

Only case 1 can be reduced to ..0101010 or ..1010101 where by reduced I mean that we don't care about the order of 01 combinations.

Firstly, ..010101 is not even on the list, secondly why can't you reduce both those to the same thing, thirdly, rubbish, of course you can't do that as then you don't have a bijection (do you even know what a bijection is?), and fouthly why aren't you answering the calls to write down the construction of the enumeration of the matrices rows? It shouldn't be very hard. Fifthly stop saying private when you mean particular, they are not synonyms.

[Organic]

Matt,

If the word induction is wrong here then we can use the word iteration instead.

[matt grime]

Originally posted by Organic
Matt,

If the word induction is wrong here then we can use the word iteration instead.

Why would that help? If you've made this construction surely you know how you made it? How come you get to be so sure of yourself and that Hurkyl and I are wrong and don't understand when you don't even know the meaning of the terms you use?

[Organic]

Matt,

..010101 is not even on the list, secondly why can't you reduce both those to the same thing, thirdly, rubbish, of course you can't do that as then you ..

Because I have the right side of my matrix then case 1 can be reduced to ...0101010 or ...1010101

[matt grime]

Originally posted by Organic
Matt,

Because I have the right side of my matrix then case 1 can be reduced to ...0101010 or ...1010101


right side of the matrix? what? What's case 1. why won't you answer the relevant questions and produce the alleged construction of the enumerated list. stop obfuscating and produce the proof of the result that *must* state why the enumerated list is in bijection with the power set, and why you are allowed to identify different elements of the power set when you are looking for a bijection!

Are you implying that the alternating sequence ...010101010 is on the right hand of the two lists you're now using? despite the fact that every row in the right hand list is eventually one 'by construction' that is the cofinite list. of course for a while you were insisting that it was on the left list too cos that was a complete enumeration of the power set as well.

[matt grime]

Originally posted by Organic
Matt,

Is it beyond you ability to understand that my matrix power_value = |N|?

Would it be beyond you to define what that sentence means? Apparently, yes.

You state the case for 2^n, then make a statement about 2^|N|. Fine but you cannot conclude anything about that case by induction or iteration from the finite case, not that you even bother to do that. In particular you cannot conclude that the information can be written encoded in a matrix with countable sets of rows and columns. One does not logically follow from the other (it is false, we have proved it).

[Organic]

Matt,

Why would that help? If you've made this construction surely you know how you made it? How come you get to be so sure of yourself and that Hurkyl and I are wrong and don't understand when you don't even know the meaning of the terms you use?

Is it beyond your ability to understand that my matrix power_value = |N|?

It is very easy because there is a bijection between the power_values and N members.

[matt grime]

So there is a bijection from the set

{2^n | n in N} and N.

So?

Or do you mean there is a bijection from a set of cardinality 2^|N| and N? Well, there isn't. Check the above post where I ask you if it is beyond you te even define what it means ot have power value |N|

so for the god knows how manyth time, write down the construction that enables you to write these matrices out and make the claims you do.

[Organic]

Matt,

Maybe this will help you:

Please tell me if you understand the set that I wrote below

3 2 1 0
2 2 2 2
^ ^ ^ ^
| | | |
v v v v
{...,1,1,1,1}
...,1,1,1,0
...,1,1,0,1
...,1,1,0,0
...,1,0,1,1
...,1,0,1,0
...,1,0,0,1
...,1,0,0,0
...,0,1,1,1
...,0,1,1,0
...,0,1,0,1
...,0,1,0,0
...,0,0,1,1
...,0,0,1,0
...,0,0,0,1
...,0,0,0,0
...


It is also can be written as: {10... ,1100... ,11110000... ,...}

[matt grime]

That does not help because you do not explain what the ellipsis of the ... means.


here is what I think it means (again, for pity's sake, why do you refuse to answer simple direct questions and observations)

we start with the two diagrams in your newdiagonal pdf that you eunmerate, the left hand one

the columns go
..0000
..0001
..0010
..0011
..0100
..0101
......
.....
.....
.....


where the first column is based on repeatin the pattern 01, the second by repeating the pattern 0011 the rth by repeating 2^r 0s then 2^r 1s (each pattern repeating infinitely many times in the column, so that you cannot claim I think it's finite, and for r goes from 1 to infinity)

now the second diagram is the first but every entry above the diagonal is a 1


you now interleave these rows alternating one from each.


Is that accurate?

[Organic]

Matt,

It is also can be written as: {10... ,1100... ,11110000... ,...}

... means that each 0 1 starting repeats on itself forever.

[matt grime]

But is my characterization equivalent to yours?

[Hurkyl]

I think he's looking at the set of columns, and assuming it's obvious that when he writes "..." he means "repeat that string of digits indefinitely"

[matt grime]

Originally posted by Organic
Matt,

It is also can be written as: {10... ,1100... ,11110000... ,...}

... means that each 0 1 starting repeats on itself forever.

the array you wrote out - clearly it satisfies every row is eventually 1 continually (ie every entry in row r is 1 after point reading right to left - the point changing as r changes) therefore it corresponds to sets whose complement is finite. The cofinite sets again. now what about them.

[matt grime]

I don't understand it. I've proved one diagram corresponds only to the finite subsets, another corresponds only to the cofinite subsets, their union is not the power set, that organic doesn't know the meaning of the terms he uses, as proven by the fact that he thinks it's ok for a bijection not to be injective. what more do i need to do?

[Hurkyl]

Matt's talking about the rows, not the columns. I think we all agree there is a bijection between the set of columns and N.

[matt grime]

Originally posted by Organic
Matt,

What is finite in {10... ,1100... ,11110000... ,...} set?



I'm not saying anything there is a finite set.


it's the same proof as before that you don't understand.

take the row at position r. let us examine what set it corresponds to in the power set.

now the c'th column starts with 2^c 1s doesn't it? that is the pattern of the columns.


the c'th entry in row r is the r'th entry in column c. Now, 2^c>c and if c>r we see the r'th entry in column c is a 1.

therefore all entries after the r'th in row r are 1.


look at the finite portion of the diagram - notice how everything above the diagonal is a 1? that's just what we've proved above.

and what that tells us is that the element in the power set correspoding to that row r contains every element in N greater than r. thus it's complement only contains some subset of the numbers 1,2,..r.

Sets like this are called cofinite.

This is the same method of proof that shows the other list only contained elements in ther power set that had a finite number of elements in them because there the any row eventually becomes zero by the same argument. thus you've only enumerated the finite and cofinite sets and none of the (uncountable) remaining sets.


The cut off point is different in each row but that doesn't matter, which is what i think you thought i meant earlier - that there was a vertical line you could draw in the diagram and all to the left of the vertical line is 0 (or 1) I am not saying that and i don't need to say that.

read the above proof carefully, and try and understand we're just formally stating that everythin above the diagonal is a 1 (in this case and 0 in the other diagram)

[matt grime]

also note i am not saying a row is finite, but that a row corresponds to a cofinite (or finite in the old case) element in the power set of N

[Organic]

Matt and Hurkyl,

What is finite in {10... ,1100... ,11110000... ,...} set?

This set is bijective to N.

Now, let us go back to look at this set like this:

3 2 1 0
2 2 2 2
^ ^ ^ ^
| | | |
v v v v
{...,1,1,1,1}
...,1,1,1,0
...,1,1,0,1
...,1,1,0,0
...,1,0,1,1
...,1,0,1,0
...,1,0,0,1
...,1,0,0,0
...,0,1,1,1
...,0,1,1,0
...,0,1,0,1
...,0,1,0,0
...,0,0,1,1
...,0,0,1,0
...,0,0,0,1
...,0,0,0,0
...

By taking the n-th notation from each member we can build a list of unique sequences which its magnitude=2^aleph0.

[matt grime]

n'th notation from what?

did you even bother to try to understand that argument? No i didn't think so.

we aren't saying the set of columns is finite. why would we? have you even read the proof i just offered? attempted to understand what it said? or did you just presume to know it was wrong? It isn't.

[Organic]

Matt, if you write this: "n'th notation from what?" it is simply shows us that you did not read carefully what I write to you and to Hurkyl, so here it is again.



{10... ,1100... ,11110000... ,...} set is bijective to N.

By taking the n-th notation from each member we can build a list of unique sequences which its magnitude=2^aleph0:


3 2 1 0
2 2 2 2
^ ^ ^ ^
| | | |
v v v v
{...,1,1,1,1}
...,1,1,1,0
...,1,1,0,1
...,1,1,0,0
...,1,0,1,1
...,1,0,1,0
...,1,0,0,1
...,1,0,0,0
...,0,1,1,1
...,0,1,1,0
...,0,1,0,1
...,0,1,0,0
...,0,0,1,1
...,0,0,1,0
...,0,0,0,1
...,0,0,0,0
...

[matt grime]

i've read and reread that post and you don't explain what the n'th notation is.

the indvidual words make sense but not when put together like that

what is a notation, what is the nth one? you didn't state what they are.

which members are you taking these notations from? members of rows, columns, power sets, N?

so illuminate it for the stupid round here


now have you actually read my proof that any row in the diagram corresponds to a confinite element of the power set?

we;ve got this diagram enumerating the cofinite sets, what's the point you're trying to make

[Organic]

Matt,

First, thank you for reading an rereading.


{10... ,1100... ,11110000... ,...} set is bijective to N.

The n-th notation of each member is:



n-th 1 2 3 . . 1 2 3 . . . . 1 2 3 . . . . . . . . 1 2 3
{1 0 . . . ,1 1 0 0 . . . ,1 1 1 1 0 0 0 0 . . . ,. . .}


By taking the n-th notation from each member we can build a list of unique sequences which its magnitude=2^aleph0:


3 2 1 0
2 2 2 2
^ ^ ^ ^
| | | |
v v v v
{...,1,1,1,1}<--> n-th 1
...,1,1,1,0 <--> n-th 2
...,1,1,0,1 <--> n-th 3
...,1,1,0,0 <--> n-th 4
...,1,0,1,1 <--> n-th 5
...,1,0,1,0 <--> n-th 6
...,1,0,0,1 <--> n-th 7
...,1,0,0,0 <--> n-th 8
...,0,1,1,1 <--> n-th 9
...,0,1,1,0 <--> n-th 10
...,0,1,0,1 <--> n-th 11
...,0,1,0,0 <--> n-th 12
...,0,0,1,1 <--> n-th 13
...,0,0,1,0 <--> n-th 14
...,0,0,0,1 <--> n-th 15
...,0,0,0,0 <--> n-th 16
...

[matt grime]

ok got it. you take the n'th element in each column as the elements in the nth row.

That does not give you "2^aleph-0" possibilities, there are exactly as many rows as there are natural numbers. How can you claim that there are 2^aleph-0 rows. Why? What set of cardinality 2^aleph-0 is it in bijection with and how?

[Organic]

Matt,

A Binary Tree with infinitely many nodes has a width of magnitude aleph0 and a length of 2^aleph0.

Simple as that.

[matt grime]

That would depend on how you defined binary tree - the rooted binary tree with countably infinite branch set has only a countable number of nodes - it is the countable union of finite sets.

Besides that has nothing to do with this issue.

Of course you don't think that binary tree means what it really means.

So what is your binary tree? You seem to think it is a Cantor set. There fore you seem to think that the rows of your matrix are in bijection with a cantor set. How? Please demonstrate? As I can prove that a cantor set is not in bijection with N, that the rows are in bijection with N, don't you think that that is a little problematic. So please exhibit this notional bijection with the Cantor set. Or explicitly state what your binary tree is. This will all involve taking (inverse/direct) limits in the correct sense, but of course you know all about that


edit: and seeing as you don't think there are an uncountable number of points in the real line, don't you think taking an uncountable subset is wrong? It is conceptually impossible in your world, and if you're in mine you need to prove all your assertions.

[Hurkyl]

Fun fact.

A (balanced) binary tree with infinitely many nodes has zero leaves.

[Organic]

Matt and Hurkyl,

Besides that has nothing to do with this issue (the Binary Tree).


Let us take again our set:

3 2 1 0
2 2 2 2
^ ^ ^ ^
| | | |
v v v v
{...,1,1,1,1}<--> n-th 1
...,1,1,1,0 <--> n-th 2
...,1,1,0,1 <--> n-th 3
...,1,1,0,0 <--> n-th 4
...,1,0,1,1 <--> n-th 5
...,1,0,1,0 <--> n-th 6
...,1,0,0,1 <--> n-th 7
...,1,0,0,0 <--> n-th 8
...,0,1,1,1 <--> n-th 9
...,0,1,1,0 <--> n-th 10
...,0,1,0,1 <--> n-th 11
...,0,1,0,0 <--> n-th 12
...,0,0,1,1 <--> n-th 13
...,0,0,1,0 <--> n-th 14
...,0,0,0,1 <--> n-th 15
...,0,0,0,0 <--> n-th 16
...

Now let us make a little redundancy diet

3 2 1 0
2 2 2 2
^ ^ ^ ^
| | | |
v v v v
... 1-1-1-1 <--> n-th 1
\ \ \0 <--> n-th 2
\ 0-1 <--> n-th 3
\ \0 <--> n-th 4
0-1-1 <--> n-th 5
\ \0 <--> n-th 6
0-1 <--> n-th 7
\0 <--> n-th 8
... 0-1-1-1 <--> n-th 9
\ \ \0 <--> n-th 10
\ 0-1 <--> n-th 11
\ \0 <--> n-th 12
0-1-1 <--> n-th 13
\ \0 <--> n-th 14
0-1 <--> n-th 15
\0 <--> n-th 16
...

and we got

3 2 1 0
2 2 2 2
^ ^ ^ ^
| | | |
v v v v
/1 <--> n-th 1
1
/ \0 <--> n-th 2
1
/\ /1 <--> n-th 3
/ 0
/ \0 <--> n-th 4
... 1
\ /1 <--> n-th 5
\ 1
\/ \0 <--> n-th 6
0
\ /1 <--> n-th 7
0
\0 <--> n-th 8

/1 <--> n-th 9
1
/ \0 <--> n-th 10
1
/\ /1 <--> n-th 11
/ 0
/ \0 <--> n-th 12
... 0
\ /1 <--> n-th 13
\ 1
\/ \0 <--> n-th 14
0
\ /1 <--> n-th 15
0
\0 <--> n-th 16
...

[matt grime]

Full credit to him, it takes some doing to talk about binary trees (uniquely path connected), even drawing them so they appear to be what you I might think, and then after much pressing claim it is actually a Cantor Set (totally disconnected), and further that as this is TD he can do whatever he damn well pleases. (Despite the fact it was your decision to move it here, Hurkyl, and that he wanted to post it in the maths forum.) That's some cheek

[matt grime]

And is that supposed to demonstrate a bijection with the Cantor set? How?

If it helps, a Cantor set is in bijection with the set of base 3 expansions of all reals between 0 and 1 with no 2 in their expansions, with the obivious dyadic representations identified.

[Organic]

Matt and Hurkyl,

3 2 1 0
2 2 2 2
^ ^ ^ ^
| | | |
v v v v
/1 <--> n-th 1
1
/ \0 <--> n-th 2
1
/\ /1 <--> n-th 3
/ 0
/ \0 <--> n-th 4
... 1
\ /1 <--> n-th 5
\ 1
\/ \0 <--> n-th 6
0
\ /1 <--> n-th 7
0
\0 <--> n-th 8

/1 <--> n-th 9
1
/ \0 <--> n-th 10
1
/\ /1 <--> n-th 11
/ 0
/ \0 <--> n-th 12
... 0
\ /1 <--> n-th 13
\ 1
\/ \0 <--> n-th 14
0
\ /1 <--> n-th 15
0
\0 <--> n-th 16
...

[Organic]

Matt,

Please look at http://www.mathacademy.com/pr/prime/articles/cantset/

Cantor set is a hybrid of base 2 and base 3 ratios.

Therefore it is easy to show that 2^aleph0 is Cantor set where Cantor set is a Binary Tree:

?
__________________________________

1 0
_____________ _____________

1 0 1 0
_____ _____ _____ _____

1 0 1 0 1 0 1 0
__ __ __ __ __ __ __ __

[matt grime]

And? That is not a bijection from the (countable) set of rows corresponding to cofinite elements the power set of N to the Cantor Set, or any thing that has cardinality 2^aleph-0


It tells you how to superimpose a binary tree onto another picture. that does not define a bijection. Nor does it imply that the binary tree is 'uncountable' (what aspect of it is uncountable in your opinion)

[matt grime]

Organic, I know what a Cantor set is.
You claimied the binary tree was a Cantor set in this thread earlier. It isn't. One wonders what you think a Cantor set is, or a binary tree.

Or why that defines a bijection.

[Organic]

Matt,

You forget my proof |R|>|N| but both N and R are countable.

[matt grime]

Evidently you don't understand the maths.

The Cantor set is not a binary tree. You have your own odd definition for binary tree, don't you. Care to explain what it is?

The cantor Set is the ***limit*** of that process.

So how does being able to draw a rooted binary tree on top of some other diagram provide you with a bijection with the Cantor set?

[Organic]

Matt,

Cantor set is a hybrid of base 2 and base 3 ratios.

Therefore Cantor set can be a base 2 fractal and base 3 fractal.

Your Cantor set is disjoint because you don't look at it from the complex level.

[Hurkyl]

Full credit to him, it takes some doing to talk about binary trees (uniquely path connected), even drawing them so they appear to be what you I might think, and then after much pressing claim it is actually a Cantor Set (totally disconnected), and further that as this is TD he can do whatever he damn well pleases. (Despite the fact it was your decision to move it here, Hurkyl, and that he wanted to post it in the maths forum.) That's some cheek

You're right. He's dropped the attitude he had for a while that caused me to start moving his posts over here in the first place.



Anyways, Organic, the trick to the "binary tree" construction of the Cantor set is that it's not enough; it only generates the points in the Cantor set with terminating decimal expansions. In order to get the whole Cantor set, you have to take the (topological) closure of this set of points.


Given a set S, the closure of S is the set of all points that are limits of sequences of elements of S. For example, the closure of (0, 1) is [0, 1]. (the sequence 0.1, 0.01, 0.001, 0.0001, ... is entirely in (0, 1), and the limit is 0, so 0 is in the closures of (0, 1))

[Organic]

Hurkyl,

You can't contradict it.

{10... ,1100... ,11110000... ,...} set is bijective to N.

The n-th notation of each member is:



n-th 1 2 3 . . 1 2 3 . . . . 1 2 3 . . . . . . . . 1 2 3
{1 0 . . . ,1 1 0 0 . . . ,1 1 1 1 0 0 0 0 . . . ,. . .}


By taking the n-th notation from each member we can build a list of unique sequences which its magnitude=2^aleph0:


3 2 1 0
2 2 2 2
^ ^ ^ ^
| | | |
v v v v
{...,1,1,1,1}<--> n-th 1
...,1,1,1,0 <--> n-th 2
...,1,1,0,1 <--> n-th 3
...,1,1,0,0 <--> n-th 4
...,1,0,1,1 <--> n-th 5
...,1,0,1,0 <--> n-th 6
...,1,0,0,1 <--> n-th 7
...,1,0,0,0 <--> n-th 8
...,0,1,1,1 <--> n-th 9
...,0,1,1,0 <--> n-th 10
...,0,1,0,1 <--> n-th 11
...,0,1,0,0 <--> n-th 12
...,0,0,1,1 <--> n-th 13
...,0,0,1,0 <--> n-th 14
...,0,0,0,1 <--> n-th 15
...,0,0,0,0 <--> n-th 16
...


If by closure you mean:

The collection of all points such that every neighborhood of these points intersects the original set in a nonempty set.

The result of this definition cannot be but at least Binary Tree where each point of it is a connaction (an intersection) with at least three neighbors (one above it and two below it).

[Hurkyl]

The problem is that each of the
10101010....
11001100....
11110000....
....

are only aleph0 long.

[Organic]

Hurkyl,

It is 2^aleph0 long because it "grows" in geometric series and not arithmetic series.


<---arithmetic series
3 2 1 0
2 2 2 2
^ ^ ^ ^
| | | |
v v v v
{...,1,1,1,1}<--> n-th 1 geometric series
...,1,1,1,0 <--> n-th 2 |
...,1,1,0,1 <--> n-th 3 |
...,1,1,0,0 <--> n-th 4 |
...,1,0,1,1 <--> n-th 5 |
...,1,0,1,0 <--> n-th 6 |
...,1,0,0,1 <--> n-th 7 |
...,1,0,0,0 <--> n-th 8 |
...,0,1,1,1 <--> n-th 9 |
...,0,1,1,0 <--> n-th 10 |
...,0,1,0,1 <--> n-th 11 |
...,0,1,0,0 <--> n-th 12 |
...,0,0,1,1 <--> n-th 13 |
...,0,0,1,0 <--> n-th 14 |
...,0,0,0,1 <--> n-th 15 |
...,0,0,0,0 <--> n-th 16 |
... V

[Hurkyl]

The problem is that the geometric growth only exists in the finite realm. You have to do some other operation (such as take a limit, or a nested union) to transfer to the infinite case, and in this case, that operation doesnt' catapault you to 2^aleph0.

[Organic]

Hurkyl,

Forget about the word "growth".

I am talking about infintley long arithmetic series = aleph0
and infinitely long geometric series = 2^aleph0.

More than that both serieses are depended.

[Hurkyl]

There is no &omega;-th term in a geometric sequence, nor in an arithmetic sequence. An infinitely long geometric (or arithmetic) sequence has aleph0 terms, one for each finite ordinal.

[Organic]

Hurkyl,

Thank you, it is just something that I used to explain Matt my point of view in some particular case.

So here it is with no -th things:

<---arithmetic series
3 2 1 0
2 2 2 2
^ ^ ^ ^
| | | |
v v v v
{...,1,1,1,1} geometric series
...,1,1,1,0 |
...,1,1,0,1 |
...,1,1,0,0 |
...,1,0,1,1 |
...,1,0,1,0 |
...,1,0,0,1 |
...,1,0,0,0 |
...,0,1,1,1 |
...,0,1,1,0 |
...,0,1,0,1 |
...,0,1,0,0 |
...,0,0,1,1 |
...,0,0,1,0 |
...,0,0,0,1 |
...,0,0,0,0 |
... V

[Hurkyl]

There is no infinite term in a geometric series. Each of the aleph0 terms are finite.

[Organic]

Hurkyl,

Now you have it in front of your eyes, two depended serieses.


The first one (arithmetic) cannot be but with aleph0 cardinality,
therefore the other one (geometric) cannot be but with 2^aleph0 cardinality.


<---arithmetic series
3 2 1 0
2 2 2 2
^ ^ ^ ^
| | | |
v v v v
{...,1,1,1,1} geometric series
...,1,1,1,0 |
...,1,1,0,1 |
...,1,1,0,0 |
...,1,0,1,1 |
...,1,0,1,0 |
...,1,0,0,1 |
...,1,0,0,0 |
...,0,1,1,1 |
...,0,1,1,0 |
...,0,1,0,1 |
...,0,1,0,0 |
...,0,0,1,1 |
...,0,0,1,0 |
...,0,0,0,1 |
...,0,0,0,0 |
... V

[Hurkyl]

First off, the rows and columns are not in an arithmetic nor a geometric sequence; they're not even numbers!

It is the width and the height that are in those sequences... however, geometric and arithmetic sequences only have finite terms. The width and height of the infinite matrix cannot be in said sequence!


Let me put it this way. Except for the first, every term in an arithmetic or geometric sequence has a term immediately before it. So if the width and height of your infinite matrix are part of these sequences, then what are the terms that come immediately before them?

[Organic]

Hurkyl,

then what are the terms that come immediately before them?

Please give some simple example in plain English.

[Hurkyl]

You're "sequences" arise from looking at pieces of the whole array. You look at a 2x1 piece, then a 4x2, then an 8x3, 16x4, ...

The thing is, each term in an arithmetic (geometric) sequence has a finite index. These sequences can only describe finite pieces of your whole array.


One of the many forms of the fundamental problem with progressing to the infinite case is this:

Each term (but the first) of an arithmetic sequence is defined to be a fixed constant plus the term before it.

You assert aleph0 is in the arithmetic sequence of widths. Well, what is the term before it?

The same complaint applies to the geometric sequence.



This fundamental problem rears its ugly head for any iterative process where each step is built from the step immediately before it. Such a process only works for terms that can be generated in a finite number of steps. There are, of course, aleph0 of these terms, but none of these terms correspond to a &omega;-th step of the process (if one even exists)

[Organic]

Hurkyl,

First off, the rows and columns are not in an arithmetic nor a geometric sequence; they're not even numbers!

If I get a list of 2^aleph0 unique elements, then their name is not important, first: they exists, second: give them a name.

This fundamental problem rears its ugly head for any iterative process where each step is built from the step immediately before it. Such a process only works for terms that can be generated in a finite number of steps. There are, of course, aleph0 of these terms, but none of these terms correspond to a ù-th step of the process (if one even exists)

Hurkyl, there is no process here but an immediate existance based on ZF axiom of infinity iterations of the power_value of the matrix:

<---arithmetic series
3 2 1 0 <---The power_value of the matrix = aleph0
2 2 2 2
^ ^ ^ ^
| | | |
v v v v
{...,1,1,1,1} geometric series
...,1,1,1,0 |
...,1,1,0,1 |
...,1,1,0,0 |
...,1,0,1,1 |
...,1,0,1,0 |
...,1,0,0,1 |
...,1,0,0,0 |
...,0,1,1,1 |
...,0,1,1,0 |
...,0,1,0,1 |
...,0,1,0,0 |
...,0,0,1,1 |
...,0,0,1,0 |
...,0,0,0,1 |
...,0,0,0,0 |
... V

[moshek]

Organic !

please do read the abstract of Haim Gaifman lecture at:

http://www.cs.tau.ac.il/~nachumd/semester.html

thank you
Moshek
[:))]

[matt grime]

How many times do we have to explain induction to you? There is a set of statements labelled by n in N, if the r'th implies the r+1st and if the 1'st is true then all the statements are ture by induction.

That is the statement is true for all n in N. It does not state the statement (ugly turn of phrase, sorry) labelled by N is true; that statement might not even make sense.


You cannot write down something labelled by N and say it is true because it is true for each n in N, that is not induction, it is plain wrong.

COUNTER EXAMPLES, again.

the set of rows in the finite diagram is finite, therefore the set of rows for N should be finite as the first set is finite, and if the n'th set is finite the n+1st set only has twice as many and is hence finite. You can't just pick and choose the properties that you think pass through



Do you ever actually think about the counter arguments to your claims or do you just blithely turn the handle and crank out another piece of 'complementary theory'?


The infinite case must be taken as a limit of something in the proper sense. You do not do this, and I am certain you do not even know how to do so. Listen to the people who do.

The 'limit' of a nested union of sets of cardinality 2^n is not 2^aleph-0. There is no substantiation for this claim other than your misuse of the axiom of infinity, which just states that the elements of N form an inductive set. |N| is not in N so you can't claim anything using it and induction.

And to make it absolutely clear, no one is saying that the power set of N does not have card 2^aleph-0, but that the list of elements you construct does not in anyway correspond to any set of that cardinality. The only reason you have for saying it is is that it follows from the finite cases - but it doesn't, otherwise it follows from the finite cases that the set of rows is finite - induction does not allow you to claim the things you are claiming *must* be true.

[Organic]

Matt,

aleph0 magnitude exists in ({},{__}) open interval where {} content is the weak limit and {__} is the strong limit.

To understand this please look at:

http://www.geocities.com/complementarytheory/Theory.pdf

Again aleph0 cannot be something which is beyond "infinitely many" because no potential infinity can be an actual infinity and the reason is very simple.

Actual infinity is the limit of any information system including Math language.

Therefore only potential infinity which defined as "infinitely many objects" can be used as an input to Math language.

Therefore the "transfinite" universes that trying to "eat the cake" (to construct a solid line by using the model of points or segments) and also to keep it untouched (to insist that this "pointed/segmented" line is beyond its points/segments) is nothing but a conceptual mistake, when the logic is Boolean logic.

In Boolean Logic x and the negation of x is false, x=solid_line AND x=infinitely many points or segments, does not hold.

[matt grime]

Hmm, looks like garbage. Seeing as you are attempting to prove things in mathematics are not sufficient your personal issues are not important - we can prove mathematics is perfectly consistent in this issue, and it is your ignorance of it that is causing you to misundertand it. Did I say aleph-0 is 'beyond n in N?' No I didn't. It isn't a natural number. Lots of things aren't natural numbers. Now, get back to the point in hand which is that you are now trying to prove that the rows of your diagram are in bijection with a Cantor set. Where is the proof of this? Their construction bijects them with N, but at no point is there a bijection witha Cantor Set. Your only indication as to why this is true is the patently false assertion that it must follow from the finite case because the rows there biject with a set of cardinality 2^n for n columns. This is not true.

[moshek]

How can you use the word Garbage to somone who creat interest of more than 24 pages on this web-site ?

Don't you feel that Organic is trying to sow you somting that you still can't recognize as new interpetation to almost everyting in mathematics?

Moshek




[:))]

[Organic]

Moshek,

It is great to see that There is onther life in Math world which is not limited to Matt's understanding abilities.


Matt,

aleph0 magnitude exists in ({},{__}) open interval where {} content is the weak limit and {__} is the strong limit.

To understand this please look at:

http://www.geocities.com/complementarytheory/Theory.pdf

Again aleph0 cannot be something which is beyond "infinitely many" because no potential infinity can be an actual infinity and the reason is very simple.

Actual infinity is the limit of any information system including Math language.

Therefore only potential infinity which defined as "infinitely many objects" can be used as an input to Math language.

Therefore the "transfinite" universes that trying to "eat the cake" (to construct a solid line by using the model of points or segments) and also to keep it untouched ( to insist that this "pointed/segmented" line is beyond its points/segments (it means a solid_line) ) is nothing but a conceptual mistake, when the logic is Boolean logic.

In Boolean Logic x and the negation of x is false, x=solid_line AND x=infinitely many points or segments, does not hold.

[moshek]

Organic:

I did not convins yet that collatz problem 3n+1 is undisideabl by the self similarity fractal type to the infinit axiom in set theory but your arguments are really beutifulls:

Please look on:

www.as.huji.ac.il/midrasha04.htm

Yours

Moshek
[:))]

[Hurkyl]

If I get a list of 2^aleph0 unique elements, then their name is not important, first: they exists, second: give them a name.

There's no good reason to believe the list has 2^aleph0 unique elements.

[Organic]

Hi Moshek,

If the result of "to be eqivanet to an axiom of some system" is true then 3n+1 is true in ZF.

This is what I write at the end of:

http://www.geocities.com/complementarytheory/3n1proof.pdf

"An axiom of some Mathematical system cannot be proved by definition.

Therefore Collatz sequences are true but cannot be proved within ZF axiomatic system."

[Organic]

Hi Hurkyl,


This fundamental problem rears its ugly head for any iterative process where each step is built from the step immediately before it. Such a process only works for terms that can be generated in a finite number of steps. There are, of course, aleph0 of these terms, but none of these terms correspond to a ù-th step of the process (if one even exists)

Hurkyl, there is no process here but an immediate existance based on ZF axiom of infinity iterations of the power_value of the matrix:

<---arithmetic series
3 2 1 0 <---The power_value of the matrix = aleph0
2 2 2 2
^ ^ ^ ^
| | | |
v v v v
{...,1,1,1,1} geometric series
...,1,1,1,0 |
...,1,1,0,1 |
...,1,1,0,0 |
...,1,0,1,1 |
...,1,0,1,0 |
...,1,0,0,1 |
...,1,0,0,0 |
...,0,1,1,1 |
...,0,1,1,0 |
...,0,1,0,1 |
...,0,1,0,0 |
...,0,0,1,1 |
...,0,0,1,0 |
...,0,0,0,1 |
...,0,0,0,0 |
... V

[Organic]

Dear Moshek Hurkyl and Matt,

I did not sleep fore more then 36 hours, so please if you want we will continue after I take some good sleep.

Bye.

[moshek]

Organic:

I really dont know yet if you are rigth, maybe.

But if you rally can show by this that the 3n+1 problem is undisidable problem in number theory, as a simple and natural Godel theorem. then it will consider as the greatest discovery in the history of the euclidian mathematics !

Good luck !
Moshek
[:))]

[Hurkyl]

Hrm, &omega; is supposed to be the omega (\omega), the symbol typically used for the ordinal that is the well-order type of the natural numbers (which is equal to the set of natural numbers in the Von Neumann model). Is it not showing up on your computer?



Ok, let's try a different tact. Let's suppose I don't understand how to make the entire array based on the sample entries you've given. Let's also suppose that I still won't get it if you give me more sample entries.

So the question is, can you explain what is your array without simply writing sample entries and presuming the reader can fill in the details?

For example, instead of saying:

.....010101

You could say

"The sequence whose n-th term is 1 if n is even and 0 if n is odd"

Instead of

.....10000001000010011

you could say

"The sequence whose n-th term is f(n) where:
f(n) := 1 if n = p^2 for some integer p
f(n) := 0 otherwise"

Instead of

.....1000100100101

you could say

"The sequence whose n-th term is f(n) where:
f(n) := 1 if n = g(p) for some integer p
f(n) := 0 otherwise
where g is recursively defined as
g(1) := 1
g(i) = g(i-1) + i for all integers i > 1"

Instead of

.....000011110000111100001111

you could say

"The sequence whose n-th term is f(n) where:
f(n) := 1 if floor(n / 4)1 is even
f(n) := 0 otherwise"

Hurkyl

1: The floor function, floor(x), \lfloor x \rfloor, is the function that "rounds down" a real number. So, floor(1) = 1, floor(2.5) = 2, floor(-3.7) = -4.

PS: It's not an entirely unreasonable supposition; you disagree with the statements I make about the array as I think it's supposed to be!

[matt grime]

I can write that Organic's musings are garbage because he does not attempt to define anything he uses and just invents some notation at random. He has claimed to have defined a multiplication on N which does not even send natural numbers to natural numbers, he makes plainly wrong statements about the axiom of infinity, he misuses the phrase binary tree, does not understand the concept of countable, thinks that mathematics implies there is a bijection from a set of cardinality aleph-0 to one of card 2^aleph-0. Very often the sentences he writes do not make sense as sentences, never mind mathematically. He also claims frequently to have a cast iron proof that is so cast iron that he has to keep rewriting it to remove errors. He has also demonstrated that he does not know what a bijection is despite claiming their use in a 'proof', he did not understand the n maps to n+1 is a bijection from N to a proper subset of itself, claiming seom bizarre things about this fact. He frequently contradicts himself and does not extend to others the courtesy he demands from them by not reading and understanding their counter arguments.

Moreover he does not seem to grasp that it is not mathematics that cannot handle infinity but his philosphy. None of the constructions and results he claims must be true are true in mathematics, and are only true in his philosophy. With these presumptions he then tells us our mathematics is faulty, when the only inconsistencies arise if we believe his unfounded assertions.

He often cites the axiom of infinity induction, yet no such thing exists without his philosophy, and it is something he refuses to explain.


One needs only to see that he believes that axioms cannot be proven true to see that he doesn't understand that which he claims is wrong - axioms are proven true trivally in an axiomatic theory.

The collatz conjecture he insisted was undecidable in ZF, despite not evidently knowing what that meant as he asserted it was equivalent to the axiom of infinity (something he was unable to prove). Of course if it is equivalent to the axiom of the infinite set then it is trivally provable in ZF. He appears now to have changed his position on that despite repeatedly berating me for being too unimaginative to see how he was correct. Another example of him contradicting himself.

Proving the Collatz conjecture is undecidable in ZF is not that big a deal really. Conway showed there are Collatz type conjectures that are undecidable. It is known the conjecture is true for all numbers less than 10^53 I believe.


If you wish to see why Organic provokes much ire from mathematicians just read all his amny posts where he passionately argues against the blindingly obvious, and cannot even understand the simple objections raised against his argument. One needs only read some of his respsonses to the challenge to prove his construction has the properties he claims to see that he doesn't understand what's going on.

[moshek]

matt : Thank you for your long answer i feel now comftable with you.
I will study cerfully your answer and replay to it tommorow.

Best

Moshek
[:))]

[matt grime]

IN particular Organic has never produced any evidence to back up his assertion that the countable set of rows in the diagram for N corresponds to a set of card 2^aleph-0. In the case for n columns the rows correspond to the power set of card 2^n, however this is not true in the infinite case, as has been clearly demonstrated. The orginal diagram had only rows with a finite number of non-zero entries, then there came a diagram interleaving with a second array where each row had only finitely many zero entries. The proofs of these facts are elementary and organic has continually refused to either accept these proofs or indicate where they are wrong. His only tactic has been to introduce yet more irrelavant things. If you wish to treat him as a mathematical thinker perhaps you, Moshek, could try and peresuade him to explain where these elementary proofs are wrong; they aren't as anyone can see, but Organic does not extend to us the courtesy he demands of others in reading and trying to understand what others write. We have read what he has written and do understand what he is attempting to do, and we can also see where he is going wrong.

[moshek]

Thank you Matt for the extension of your view and your really deep care for mathematics that teach me. I promise you to do all the best I can do for understanding of the situation were we are now !!

sincerely
Moshek

[:))]

[Organic]

Matt,

We have read what he has written and do understand what he is attempting to do...

No Matt you you do not understand what I am doing because my point of view is a paradigm change of the infinity concept used by Math language.

There are two kinds of it:

1) Actual infinity (the word "many" cannot be used) which is the unreachable limit of Math language.

2) Potential infinity (infinitely many objects) which is the only form
that can be used by mathematics language.

If you don't distinguish between these two forms of infinity concept, then you cannot say even a single word about my mathematical definitions.

Simple as that.


And for you Moshek, it looks that you have no opinion of your own and you are trying to be nice to everyone.

So, please let me tell you, this is not the way, you have to choose your side.

[moshek]

Well Organic you know already that I am on the side were you are since I work on this discovery by myself for the last 20 years. This is not the point here . I try to understand matt as much as I can to find some bridge among the gap now. you must understand that the fact that you don’t have many background in the tradition mathematics make it very difficult to understand you. and they may say by a mistake that it is a Garbage. but they are really care from there heart on mathematics.

Moshek
[:((]

[Organic]

Dear Moshek,[:)]

In this case you have to show clearly what is your fundamental paradigm about the infinity concept and also clearly show what do you think we have to do if we want to show to the people of the old paradigm, how they can make Math under the new paradigm.

Can you do that?

[matt grime]

As I've said before you are prefectly entitled to whatever philosophical position you want. What I will not accept is you telling us that mathematics cannot cope with something that it makes no claims to explain, and is not mathematical. The only problems arise when we accept your ideas, ideas which are inconsistent and ill-defined. If you want to show mathematics is inconsistent you cannot work from outside, you must demonstrate it internally.

Noticably you don't defend yourself against the allegations that you do not read or understand other people's posts.

It is only your opinion that the rows you write down are in correspondence with a set of cad 2^aleph-0. We can prove it isn't as the property of being in bijection is an elementary equivalence relation, only in your theory is there a problem with this; it is only if one declares that the words 'many' and 'for all' are incompatible with logic that your problems with cantor arises, however that isn't mathematics.

It is now clear that you cannot refute the disproof that the rows are not in correspondence with the power set, which shows how poor your abstract thought is.

Do you want another thing to talk about?

You've demanded we provide a multiplication on N that is complementary to addition.

Let's do that.

1. What do you mean by complementary

2. What is a multiplication?


Give the rules this notional operation must define.

Note yours

1. Is not a binary operation from NxN to N

2. Does not associate,

3. Either does not distribute over addition, or have 1 as the identity (if it had both then it would be commutative).

So what must something be to qualify as a 'multiplication' and be complementary to addition.

(And you claim we don't read and understand you)

[matt grime]

Originally posted by Organic
Hurkyl,

Can you please show us how we can use "even" and/or "odd" when there are infinitely many '1' notations and infinitely many '0' notations in some 01 infinitely long sequence?

If you ask me then if there is a way to define 01 infinitely long sequence where "even" and/or "odd" cannot be defined, then this sequence is bijective to N members.

And this is exactly what I mean when I speak about aleph0.

In aleph0 magnitude "even" and/or "odd" can't be used.


It's very simple, Organic, and if you understood this your life would be so much better. Although there are 'aleph-0' entries, each entry is either in an odd or even position (at a finite point in the sequence), therefore it makes perfect sense to say the string

(x_i) i in N is 1 when i is even 0 when i is odd.

Aleph-0 is not a number, there is not an aleph-0'th entry in the sequence, every element in the sequence occurs at some n in N, which is either an even or an odd number.

[moshek]

Dear Matt !!

I am afraid that i will not be able to answer you today i am writing a paper on 01 laws and i have to finish it untill thursday.

But i will certanly do that in few days.

take care
Moshek
[t)]

[moshek]

and also to you Dear Organic

Sorry

Moshek
[:)]

[Organic]

Hurkyl,

For example, instead of saying:

.....010101

You could say

"The sequence whose n-th term is 1 if n is even and 0 if n is odd"

Can you please show us how we can use "even" and/or "odd" when there are infinitely many '1' notations and infinitely many '0' notations in some 01 infinitely long sequence?

If you ask me then if there is a way to define 01 infinitely long sequence where "even" and/or "odd" cannot be defined, then this sequence is bijective to N members.

And this is exactly what I mean when I speak about aleph0.

In aleph0 magnitude "even" and/or "odd" can't be used.

If we examine again the structure of the most right column of the matrix which is drawn below we find that the length of the list has at least aleph0 magnitude.

<---arithmetic series
3 2 1 0 <---The power_value of the matrix = aleph0
2 2 2 2
^ ^ ^ ^
| | | |
v v v v
{...,1,1,1,1} geometric series
...,1,1,1,0 |
...,1,1,0,1 |
...,1,1,0,0 |
...,1,0,1,1 |
...,1,0,1,0 |
...,1,0,0,1 |
...,1,0,0,0 |
...,0,1,1,1 |
...,0,1,1,0 |
...,0,1,0,1 |
...,0,1,0,0 |
...,0,0,1,1 |
...,0,0,1,0 |
...,0,0,0,1 |
...,0,0,0,0 |
... V

Now I am going to show my new point of view about the potential infinity.

First let us look at this list:

<---arithmetic series
3 2 1 0 <---The power_value of the matrix = aleph0
2 2 2 2
^ ^ ^ ^
| | | |
v v v v
{... 1 1 1 1} geometric series
... 1 1 1 |
... 1 1 1 |
... 1 1 / |
... 1 1 1 |
... 1 1 / |
... 1 //1 |
... 1 // / |
... 1 1|1 |
... 1 1| |
... 1 ||1 |
... 1 // |
... /|1 1 |
... /|1 |
... / || 1 |
... / || |
... 1 || V

the same can be done with '0' notations

Shotly speaking the main structure here is:

<---arithmetic series
3 2 1 0 <---The power_value of the matrix = aleph0
2 2 2 2
^ ^ ^ ^
| | | |
v v v v
{... 1 1 1 1} geometric series
... 1 1 1 |
... 1 1 |
... 1 1 |
... 1 |
... 1 |
... 1 |
... 1 |
... |
... |
... |
... |
... |
... |
... |
... |
... V

again, the same can be done with '0' notations

Now we shall show thet this information is greather then aleph0 magnitude.

Step 1: we will show again our list in this way:

<---arithmetic series
3 2 1 0 <---The power_value of the matrix = aleph0
2 2 2 2
^ ^ ^ ^
| | | |
v v v v
{...,1,1,1,1} geometric series
...,1,1,1,0 |
...,1,1,0, |
...,1,1,0, |
...,1,0, , |
...,1,0, , |
...,1,0, , |
...,1,0, , |
...,0, , , |
...,0, , , |
...,0, , , |
...,0, , , |
...,0, , , |
...,0, , , |
...,0, , , |
...,0, , , |
... V

step 2: to make it clearer we shall show it now in this way:

<---arithmetic series
3 2 1 0 <---The power_value of the matrix = aleph0
2 2 2 2
^ ^ ^ ^
| | | |
v v v v
{...,1,1,1,1} <--> 1 geometric series
...,1,1,1, |
...,1,1, , |
...,1,1, , |
...,1, , , |
...,1, , , |
...,1, , , |
...,1, , , |
... |
...,0,0,0,0 <--> 2 |
...,0,0,0, |
...,0,0, , |
...,0,0, , |
...,0, , , |
...,0, , , |
...,0, , , |
...,0, , , |
... |
...,1,1,1,1 <--> 3 |
...,1,1,1, |
...,1,1, , |
...,1,1, , |
...,1, , , |
...,1, , , |
...,1, , , |
...,1, , , |
... |
...,0,0,0,0 <--> 4 |
...,0,0,0, |
...,0,0, , |
...,0,0, , |
...,0, , , |
...,0, , , |
...,0, , , |
...,0, , , |
... |
...,1,1,1,1 <--> 5 |
...,1,1,1, |
...,1,1, , |
...,1,1, , |
...,1, , , |
...,1, , , |
...,1, , , |
...,1, , , |
|
... V

As we can simply and clearly see, there is no bijection between the natural numbers and the infinitely long geometric series of my list.

Therefore we can conclude that the length magnitude of our list > aleph0 magnitude.

Please pay attention that in this case ...0101 most right column length has 2^aleph0 magnitude.

Therefore the width and length of my 01 matrix are both countable.

3 2 1 0 3 2 1 0
2 2 2 2 2 2 2 2
^ ^ ^ ^ ^ ^ ^ ^
| | | | | | | |
v v v v v v v v
...0 0 0 0 <--> 1 or ...1 1 1 1 <--> 1
...0 0 0 1 <--> 2 or ...1 1 1 0 <--> 2
...0 0 1 0 <--> 3 or ...1 1 0 1 <--> 3
...0 0 1 1 <--> 4 or ...1 1 0 0 <--> 4
...0 1 0 0 <--> 5 or ...1 0 1 1 <--> 5
...0 1 0 1 <--> 6 or ...1 0 1 0 <--> 6
...0 1 1 0 <--> 7 or ...1 0 0 1 <--> 7
...0 1 1 1 <--> 8 or ...1 0 0 0 <--> 8
...1 0 0 0 <--> 9 or ...0 1 1 1 <--> 9
...1 0 0 1 <--> 10 or ...0 1 1 0 <--> 10
...1 0 1 0 <--> 11 or ...0 1 0 1 <--> 11
...1 0 1 1 <--> 12 or ...0 1 0 0 <--> 12
...1 1 0 0 <--> 13 or ...0 0 1 1 <--> 13
...1 1 0 1 <--> 14 or ...0 0 1 0 <--> 14
...1 1 1 0 <--> 15 or ...0 0 0 1 <--> 15
...1 1 1 1 <--> 16 or ...0 0 0 0 <--> 16
... or ...

or a mixing of them

[matt grime]

your view on infinity means that the rows are simultaneously countable, and that they are NOT in bijection with N. Thus your theory proves a statement that is simultaneous true and false. And you think our is the faulty system?

[Organic]

No Matt,

It is true and false iff countability is related to N (as it understood by standard Math).

My point of view clearly shows that this is not the case, and standard Math does not understand the infinity concept.

[Hurkyl]

As we can simply and clearly see, there is no bijection between the natural numbers and the infinitely long geometric series of my list.

It looks to me like there's one block of length 8 for every natural number. (and nothing else)

Thus, I can make a list such that there's one row for each natural number (and nothing else) through the following mapping:

For any natural number n, write it as 8p + q for p and q integers and 1 < q <= 8. Then, I map n with the q-th row in the p-th block.


3 2 1 0 3 2 1 0
2 2 2 2 2 2 2 2
^ ^ ^ ^ ^ ^ ^ ^
| | | | | | | |
v v v v v v v v
...0 0 0 0 <--> 1 or ...1 1 1 1 <--> 1
...0 0 0 1 <--> 2 or ...1 1 1 0 <--> 2
...0 0 1 0 <--> 3 or ...1 1 0 1 <--> 3
...0 0 1 1 <--> 4 or ...1 1 0 0 <--> 4
...0 1 0 0 <--> 5 or ...1 0 1 1 <--> 5
...0 1 0 1 <--> 6 or ...1 0 1 0 <--> 6
...0 1 1 0 <--> 7 or ...1 0 0 1 <--> 7
...0 1 1 1 <--> 8 or ...1 0 0 0 <--> 8
...1 0 0 0 <--> 9 or ...0 1 1 1 <--> 9
...1 0 0 1 <--> 10 or ...0 1 1 0 <--> 10
...1 0 1 0 <--> 11 or ...0 1 0 1 <--> 11
...1 0 1 1 <--> 12 or ...0 1 0 0 <--> 12
...1 1 0 0 <--> 13 or ...0 0 1 1 <--> 13
...1 1 0 1 <--> 14 or ...0 0 1 0 <--> 14
...1 1 1 0 <--> 15 or ...0 0 0 1 <--> 15
...1 1 1 1 <--> 16 or ...0 0 0 0 <--> 16
... or ...


or a mixing of them

Both of these are countable (you've even "counted" them!), and so is any "mix" of these.

[Hurkyl]

It is true and false iff countability is related to N.

And countability is related to N by definition.

[Organic]

Hurkyl,

It is true and false iff countability is related to N (as it understood by standard Math).

Standard math language is a "quntitative orianted system"

My point of view is first of all "structural orianted", therefore I can see standatd math universe But standard Math has no ability to my univerese.

Shortly speaking , in my univerese there is information That standard math ignore.

"STRUCTURAL ORIANTED" THEORY OF NUMBERS IS REACHER THAN "QUANTITATIVE ORIANTED" THEORY OF NUMBERS(= the current standard Math language).

[Organic]

Hurkyl,

You did not understand my example so, I'll write is and explain it again:


<---arithmetic series
3 2 1 0 <---The power_value of the matrix = aleph0
2 2 2 2
^ ^ ^ ^
| | | |
v v v v
{...,1,1,1,1} <--> 1 geometric series
...,1,1,1, |
...,1,1, , |
...,1,1, , |
...,1, , , |
...,1, , , |
...,1, , , |
...,1, , , |
... |
...,0,0,0,0 <--> 2 |
...,0,0,0, |
...,0,0, , |
...,0,0, , |
...,0, , , |
...,0, , , |
...,0, , , |
...,0, , , |
... |
...,1,1,1,1 <--> 3 |
...,1,1,1, |
...,1,1, , |
...,1,1, , |
...,1, , , |
...,1, , , |
...,1, , , |
...,1, , , |
... |
...,0,0,0,0 <--> 4 |
...,0,0,0, |
...,0,0, , |
...,0,0, , |
...,0, , , |
...,0, , , |
...,0, , , |
...,0, , , |
... |
...,1,1,1,1 <--> 5 |
...,1,1,1, |
...,1,1, , |
...,1,1, , |
...,1, , , |
...,1, , , |
...,1, , , |
...,1, , , |
|
... V

Standard mapping looks at the block as if it is a one element, but each block has an extra information that gives to the 0 1 sequences the magnitude of 2^aleph0 in this list.

Shortly speaking, when we deal with infinitely many elements the structural information of our data cannot be ignored, and it has a tremendous impact on the results.

In this case, the same ...10101 sequence can have aleph0 magnitude or greather.

Shortly speaking through my more sensative "structural orianted" point of view we can write this:

aleph0 < 2^aleph0 < 3^aleph0 < 4^aleph0 < ...

aleph0 < aleph0+1

As you can see, my aleph0 is not your aleph0 because your aleph0 does not exist, and the reason is: Standard Math system does not distinguish between actual infinity and potential infinity, as I clearly and simply show here:

http://www.geocities.com/complementarytheory/RiemannsLimits.pdf

Again no collection of infinitely many elements can be completed.

Therefore the +1 in alpeh0+1 is always beyond the scope of aleph0, which means: there is no bijection between a collection with aleph0 magnitude and a collection with apeh0+1 magnitude.

Simple as that.

[matt grime]

Originally posted by Organic
No Matt,

It is true and false iff countability is related to N (as it understood by standard Math).

My point of view clearly shows that this is not the case, and standard Math does not understand the infinity concept.

So you want to claim this is a contradictin in ordinary maths? Well, then you must prove that the enumerated rows are in bijection with a set of cardinality 2^aleph-0 using our mathematics. Since in our mathematics bijectivity is an equivlaence relation, this cannot be true, and therefore something in your logic has gone wrong. This is easily spotted at the point where you claim a geometric series has 2^aleph-0 elements. That is not true, we can prove it is not true. It is only your opinion that it is true. If it is not just an opinion prove it. You can't as you'd have to demonstrate that the notion of bijection is wrong in our mathematics,.

[Organic]

Matt,

Please read the last two posts that I wrote to Hurkyl.

Thank you.

[matt grime]

I did read your last posts to Hurkyl. They are unclear and suffer from not having enough words explaining them. However they are also self-contradictory as by enumerating them by N you are providing a bijection with N. You are then claiming a bijectio with 2^aleph-0 if we can abuse notation like that. As I can prove that there is no bijection from N to P(N) then you've made a mistake somewhere. The proof is valid in mathematics even if you don't like it. We can use the words 'for all' and 'many'.

Why when you generate a contradiction do you assume the entireity of mathematics is wrong and not even consider the option that perhaps you, who has no formal mathematical training and appears to have never read a mathematics text book (beyond high-school) could have misunderstood something?


There is nothing in your posts which explains why the cardinality of a geometric sequence is 2^aleph-0. I imagine this will provoke the response that I have a closed mind unwilling to be imaginative. However, as you are attempting to make this claim be true in my interpretation of mathematics you have to write it mathematically, not in your personal language.

So, provide some compelling evidence such as a proof that the set of rows in your diagram has cardinality 2^aleph-0.

[Organic]

Dear Matt,

I did read your last posts to Hurkyl. They are unclear and suffer from not having enough words explaining them. However they are also self-contradictory as by enumerating them by N you are providing a bijection with N. You are then claiming a bijectio with 2^aleph-0 if we can abuse notation like that. As I can prove that there is no bijection from N to P(N) then you've made a mistake somewhere. The proof is valid in mathematics even if you don't like it. We can use the words 'for all' and 'many'.


Please read this again:

In this case, the same ...10101 sequence can have aleph0 magnitude or greather.

Shortly speaking through my more sensative "structural orianted" point of view we can write this:

aleph0 < 2^aleph0 < 3^aleph0 < 4^aleph0 < ...

aleph0 < aleph0+1

As you can see, my aleph0 is not your aleph0 because your aleph0 does not exist, and the reason is: Standard Math system does not distinguish between actual infinity and potential infinity, as I clearly and simply show here:

http://www.geocities.com/complementarytheory/RiemannsLimits.pdf

Again no collection of infinitely many elements can be completed.

Therefore the +1 in alpeh0+1 is always beyond the scope of aleph0, which means: there is no bijection between a collection with aleph0 magnitude and a collection with apeh0+1 magnitude.

Simple as that.

[Organic]

Hurkyl,


It looks to me like there's one block of length 8 for every natural number. (and nothing else)

No it has infinitley many finite collections of '0' or '1' notations for each natural number, that are ignored by Standard Math mapping.

[matt grime]

So in order to create a problem with mathematics you see fit to redefine its terms to mean something different.

That does not prove anything. You cannot prove something is incorrect if you are not using it properly.

It is only possible to say that there is a mistake if ou stick within the system.

So try and prove the error without redefining terms to suit your needs. Using only the correct mathematics prove mathematics is not consistent. That is your challenge, that is what you've been claiming is going on.

[Organic]

Dear Matt,

That does not prove anything. You cannot prove something is incorrect if you are not using it properly.

By Standard Math aleph0+1 = aleph0 because Standard Math is first of all a "quantitative oriented" system.

My Math is first of all a "structural oriented" system.

Here is some simple example:

By my Math system there is a structural difference between a solid line and a collection of segments that have the same length of this solid line.

Shortly speaking no collection of segments can be a solid line.

They can have the same length, but this is only the quantitative point of view of this comparison.

My point of view is looking for both structural and/or quantitative properties before it air her view about some conclusion.

The structure of a solid line is what I call “an actual infinity”.

This structure cannot be reached by any collection of segments, for example: no collection of glass’s broken pieces can construct a smooth “one piece” glass.

My Math word products are the complementary association between these different structural states.

If we use some analogy then my Math system is the associations between the collection of glass’s broken pieces AND the smooth “one piece” glass.

Therefore my natural number is first of all examined by its structural properties, where quantity is the invariant state which is being kept when structural property is changed (by this way we have more information, which is unreachable by quntitative point of view).

The result of this point of view define the natural numbers not only by their quantitative property but also by their internal structural changes when some quantity is given.

If you understand this fundamental point of view on Math language, then please look again at this example:


A set is only a framework to explore our ideas.

The concept of an oredered set does not depend on the quantity concept as shown here:

By Complementary Logic multiplication is noncommutative,
but another interesting result is the fact that multiplication
and addition are complementary opreations that can be ordered
by different symmetry degrees where quantity remains unchanged
for example:

A Number is anything that exists in ({},{__})

Or in more formal definition:

({},{_}):={x|{} <-- x(={.}) AND x(={._.})--> {_}}

Where -->(or <--) is ASPIRATING(= approaching, but cannot become closer to).

If x=4 then number 4 example is:

Number 4 is a fading transition between multiplication 1*4 and
addition ((((+1)+1)+1)+1) ,and vice versa.

This fading can be represented as:


(1*4) ={1,1,1,1} <------------- Maximum symmetry-degree,
((1*2)+1*2) ={{1,1},1,1} Minimum information's
(((+1)+1)+1*2) ={{{1},1},1,1} clarity-degree
((1*2)+(1*2)) ={{1,1},{1,1}} (no uniqueness)
(((+1)+1)+(1*2)) ={{{1},1},{1,1}}
(((+1)+1)+((+1)+1))={{{1},1},{{1},1}}
((1*3)+1) ={{1,1,1},1}
(((1*2)+1)+1) ={{{1,1},1},1}
((((+1)+1)+1)+1) ={{{{1},1},1},1} <------ Minimum symmetry-degree,
Maximum information's
clarity-degree
(uniqueness)


============>>>

Uncertainty
<-Redundancy->^
3 3 3 3 | 3 3 3 3
2 2 2 2 | 2 2 2 2
1 1 1 1 | 1 1 1 1 1 1 1 1 1 1
{0, 0, 0, 0} V {0, 0, 0, 0} {0, 1, 0, 0} {0, 0, 0, 0}
. . . . . . . . . . . . . . . .
| | | | | | | | | | | | | | | |
| | | | |__|_ | | |__| | | |__|_ |__|_
| | | | | | | | | | | |
| | | | | | | | | | | |
| | | | | | | | | | | |
|__|__|__|_ |_____|__|_ |_____|__|_ |_____|____
| | | |
(1*4) ((1*2)+1*2) (((+1)+1)+1*2) ((1*2)+(1*2))

4 = 2 2 2
1 1 1 1 1 1 1
{0, 1, 0, 0} {0, 1, 0, 1} {0, 0, 0, 3} {0, 0, 2, 3}
. . . . . . . . . . . . . . . .
| | | | | | | | | | | | | | | |
|__| |__|_ |__| |__| | | | | |__|_ | |
| | | | | | | | | | |
| | | | |__|__|_ | |_____| |
| | | | | | | |
|_____|____ |_____|____ |________| |________|
| | | |
(((+1)+1)+(1*2)) (((+1)+1)+((+1)+1)) ((1*3)+1) (((1*2)+1)+1)

{0, 1, 2, 3}
. . . .
| | | |
|__| | |
| | | <--(Standard Math language uses only this no-redundancy_no-uncertainty_symmetry)
|_____| |
| |
|________|
|
((((+1)+1)+1)+1)


Multiplication can be operated only among objects with structural identity,
where addition can be operated among identical and non-identical
(by structure) objects.

Also multiplication is noncommutative, for example:

2*3 = ( (1,1),(1,1),(1,1) ) , ( ((1),1),((1),1),((1),1) )

3*2 = ( (1,1,1),(1,1,1) ) , ( ((1,1),1),((1,1),1) ) , ( (((1),1),1),(((1),1),1) )

[matt grime]

So does this mean that you now accept that there is no problem in Cantor's Diagonal argument in the extant mathematical world where aleph-0 and 2^aleph-0 are clearly and demonstrably not the same, and that the constructions you claim must be true are not in fact constructions that hold in mathematics as is held to be true today? In short do you accept that there is no contradiction in the current mathematics which uses 'infinite' and 'many' and 'for all'?

If not then explain why it is inconsistent within itself. Do not mention you're structural view point, do not mention your definition of any objects, unless they are things that current mathematics holds to be valid.


Further, reread the posts where I ask you to explain in what sense your 'multiplication is a 'multiplication' - it is not even a binary operation from NxN to N, is it?

[Organic]

Matt,

First let us see what the meaning of a non-empty set is.

A non-empty set of natural numbers does not depend on the quantity of the numbers which included in it.

All we have to know is that each number which included in it fits to the definition of the natural numbers.

The definition is important and not the set that contain its products.

Now, there is one definition and infinitely many products which we call natural numbers.

We define some set, called N, that contains any product that can be defined by the natural numbers definition.

Now we want to know how many products included in N or in other words, we want to know the cardinality of N.

The value of a cardinal is the answer to the question "How many?".

So, what is the meaning of the cardinal concept when we have infinitely many elements?

It cannot be a quantity because we cannot ask "how many?" when we deal with infinitely many elements.

So there are two solutions to this condition.

Solution 1) We can change the basic meaning of cardinality, by saying that the cardinality of infinitely many elements does not measured by quantity but by the magnitude (or size) of the collection.

By choosing this way we define two different meanings to cardinality in Math language that cannot agree with each other.

More than that, by using concepts like magnitude or size we actually saying that there is a way summarize infinitely many elements.

But there is a fundamental conceptual problem here which is: The very nature of infinitely many elements is that they cannot be summarized, because no collection of infinitely many elements is a complete collection.

So there are to big problems two solution 1, which are:

a) A main concept like cardinality has two different meanings that do not agree with each other.

b) By summarizing infinitely many elements we are going against the very nature of infinitely many elements which is: they cannot be summarized, because no collection of infinitely many elements is a complete collection by definition.


Solution 2) We do not change the meaning of cardinality, but the clarity of its result, when we deal with infinitely many elements, which means: when we have a collection of infinitely elements quantity is unknown, then our only well known information is that any element that included in this collection, fits
to the definition of the natural numbers.

Shortly speaking, by using ", ..." notation we clearly saying that we accept the vogue nature if cardinality, when we deal with a collection of infinitely many elements like the set of Natural numbers, which is notated in this way:
{1, 2, 3, ...}.

By this attitude we do not force any artificial summa on infinitely many elements and accept their nature of not being completed.

Cantor chose solution 1, and the result is two separated worlds (finite and infinite) in today's Math.

Also by the transfinite word aleph0+1 = aleph0, 2^aleph0 = 3^aleph0 (and so on) which means that Cantor's attitude has no ability to distinguish between (for example) aleph0+1 and aleph0, and by this result we can understand that this mathematical system looses information (that can be important) on each step of it.

On the contrary solution 2 can distinguish between aleph0+1 and aleph0 because through its point of view aleph0+1 > aleph0 and also 3^aleph0 > 2^aleph0 (and so on) and can use this differences as a valuable information to make Math.

My Math is based on solution 2 that can be used to develop a much more sensitive and interesting systems then solution 1.

[matt grime]

Firstly, the cardinals do not need to come with any innate ring structure such as addition - we choose to define one which has certain unintuitive (to some) properties, all of which summarize the simple idea that if S is a set that is not finite, then then S can be put in bijection with plenty of other sets such as Su{1}, Su{1,2},..,SuS etc.

There are symbols (that can be) associated to sets that aren't finite called ordinals, w is the first infinite ordinal and has the property that w is not equal to w+1, because of the nature of order preservation required.

All the things you allude to have
already been defined properly.

A perfect example of these is realized in the (well ordered) sets

{k-1/(n+1) | n,k in N}

with the element r-1/(s+1) corresponding to the ordinal rw+s (there is a small error here in the exact labelling, I hope you will allow that this is just supposed to be an indicator of this area)

There are therefore infinitely many well ordered, distinct sets with distinct ordinals that have the same cardinality.

I'm sorry that you cannot use the word complete for a set that is not finite, no one else has a problem with N being the complete set of Natural numbers, and it not being finite. Perhaps you ought to think that your interpretation of the word 'complete' is at fault?

Notice that I've only used the word infinite once in that, and that was as part of a noun, and at no point did I have to even use the word infinity.

When a mathematician speaks of cardinals measuring 'quantity' or size of sets, it is a generalization of the finite case, such generalizations do not share all of the properties of the original case, but they are useful, and provide us with lots of tools, such as the ability to prove that transcendental numbers exist without having to construct them.

Cardinality does not have two different meanings, it has one.

[Organic]

Matt,

Cardinality does not have two different meanings, it has one.

So please tell me exactly how many n's there are in N.

Also ordinals has to be distinguished from each other before we can use them, so redundancy and uncertainty are not allowed.

[matt grime]

Asking me to state 'exactly how many ns there are in N' is an ill-founded question. Where have I said there are X many natural numbers (or elements of an infinte set) in anything other vein than an attempt to communicate in your language or illuminate a point?

I have attempted whenever I am trying to be rigorous to switch to more correct langauge and use the phrase 'the set of rows has card aleph-0' rather than use your abuse of notation and say 'there are aleph-0 many rows' though I have often for expediency abused the notation in the same way. The word 'many' in such instances is not being used in a rigorous way.

The natural numbers have cardinality aleph-0. Aleph-0 tells us what the isomorphism type of the set is. Two sets have the same cardinality iff they are isomorphic. This would be lost in your definition. Cardinality is about the set-maps between sets, that's all.

www.dpmms.cam.ac.uk/~wtg10/ordinals.html

see what a fields medallist thinks.

[Organic]

Matt,

Two sets have the same cardinality iff they are isomorphic. This would be lost in your definition

By my definition I can distinguish between aleph0+1 and aleph0.

By your definition you cannot distinguish between them.

So, where is the advantage of your system, that we have to keep?

[matt grime]

But what is your definition of aleph-0 and aleph-0 + 1?

Two sets with an isomorphism between them no longer have the same cardinality

Now suppose S and T are sets and there is an injection from S to T. Does your system imply |S|<|T|?

[Organic]

Matt,

Because by my point of view the cardinality of infinitely many elements is unknown (no collection of infinitely many elements is completed) I use aleph0 as the notation of this open (non completed) state.

By doing this I can use any operation that we use between finite collections for example aleph0/2 < aleph0.

For example: the number of odd numbers in N is aleph0/2.

[matt grime]

But then what is the cardinality of an infinite set that isn't a set of natural numbers, or a subset of them?

For instance, what is the cardinality of the set of finite groups, what is the cardinality of the set of algebraic integers, what is the cardinality of the set of functions from projective n space to projective m space? What is the cardinality of the underlying field of the rank one free module of D_2n over an algebraically closed field of characteristic 2?

There is more to life than just the natural numbers. Unless you about to invent a different cardinal for every set you ain't getting much.

And I can differentiate between the set of integers and the set of even integers because they are different (but isomorphic) sets.

[Organic]

Matt,

Aleph0 is only the basis of a collection of infinitely many elements.

For example 2^aleph0 = |R|

For example (that you already know):


<---arithmetic series
3 2 1 0 <---The power_value of the matrix = aleph0
2 2 2 2
^ ^ ^ ^
| | | |
v v v v
{...,1,1,1,1} geometric series
...,1,1,1,0 |
...,1,1,0,1 |
...,1,1,0,0 |
...,1,0,1,1 |
...,1,0,1,0 |
...,1,0,0,1 |
...,1,0,0,0 |
...,0,1,1,1 |
...,0,1,1,0 |
...,0,1,0,1 |
...,0,1,0,0 |
...,0,0,1,1 |
...,0,0,1,0 |
...,0,0,0,1 |
...,0,0,0,0 |
... V


Also please pay attention that 3^aleph0 > 2^aleph0

[matt grime]

But why is the card of R 2^aleph-0?

Every real number has an integer part, and there are aleph-0 of those, for each of these surely there are 10^aleph-0 possible decimal expansions of the non-integer part? so |R| is aleph-0*10^aleph-0. But wait what if I chose base 2,3,4 or 7 expansions, surely then |R| is aleph-0*3^aleph-0, so they must be the same, cancelling aleph-0 surely 2^aleph-0=3^aleph-0....

so, what's your justification for defining |R|=2^aleph-0

you said any cosntruction I could do with finite sets I could do with infinite ones, so I just did.


So, what's the cardinality of the set of all finite groups?

[matt grime]

I've got an even better one.

What's the cardinality of the rationals?

[Hurkyl]

By my definition I can distinguish between aleph0+1 and aleph0.

By your definition you cannot distinguish between them.

So, where is the advantage of your system, that we have to keep?


There is an (essentially) unique statistic about sets that tells us when sets have bijections between each other.

My definition of cardinality is that statistic.

Whatever sort of thing aleph0 is supposed to be, it is not that statistic.

Therefore, your definition is entirely useless if I want to know a statistic that tells us when sets have bijections between each other, however my definition works.

[Organic]

Matt,

so, what's your justification for defining |R|=2^aleph-0

My mistake, I mean |R|=base_value>1^aleph0.

surely 2^aleph-0=3^aleph-0....

surely 2^aleph-0<3^aleph-0....

because in 3 notations we have much more combinations in both width and length of the 0,1,2 matrix, then in the width and length of the 0,1 matrix.

Shortly speaking, my system does not ignore these differences between 2^aleph0 and 3^aleph0 matrixes and can use them to make math.

Your system cannot do that.

What's the cardinality of the rationals?

Please look at page 5 in this paper:

http://www.geocities.com/complementarytheory/NewDiagonalView.pdf

[Organic]

Hurkyl,
There is an (essentially) unique statistic about sets that tells us when sets have bijections between each other.

My definition of cardinality is that statistic.

Whatever sort of thing aleph0 is supposed to be, it is not that statistic.

Therefore, your definition is entirely useless if I want to know a statistic that tells us when sets have bijections between each other, however my definition works.

So, all you have is: there is(=1) or there is not(=0) a bijection between two sets.

But I can be much more sensative then you and find an interesting information that exists between your 0 1 statistic results, for example:

2^aleph0 < 3^aleph0 because base value of 3 creates more information (that can be explored) then base 2.

The same is about aleph0 < 2^aleph0 or aleph0+1 > aleph0.

The finite part of each operation here can be used to give us statistical results, which are much more interesting then any general statistics, which is reduced to and based on 0 1 results.

Your definitions work, but what a price you pay (by ignoring information that can be very important)?

Shortly speaking, your word is a synthetic 0 1 digital world.

My world is an analogical world that can use details and also can give statistical results, which are much more accurate then your 0 1 digital statistical world.

For example, from my point of view the CH problem is an artificial problem that was forced on infinitely many elements that can be understood only by 0 1 digital point of view.

Shortly speaking, from my analogical point of view Cantor's world is nothing but a collection of shortcuts that do not distinguish between the simple and the trivial.

Let me give you some example:

We have this inifintely long periodic patterns list:

0
0
1
0
0
1
0
0
1
.
.
.

Let us say that we want to know the ratio between notations '1' and '0' in this infinitely long list.

By your system we can find a bijection between '1' and '0' notations:

0 <--> 1
0 <--> 1
0 <--> 1
...

So the ratio value does not exist when we deal with infinitely many elements (and it cannot be used to make Math)


By my system the ratio r1=aleph0/3 or aleph0*(1/3), the ratio of r0=aleph0*(2/3) and r1+r0=aleph0.

There is a conceptual problem in the basis of the "transfinite" world.

By using the word "transfinite" we mean that we can capture the all collection of infinitely many elements where the capturer tool does not belong (transcendent) to the elements which it captures.

R set is a complete set (no gaps) that described as “given any arbitrary interval, this interval includes infinitely many points which are connected to each other”.

Now please show me how infinitely many elements can be both unique AND non-unique (connected) on the same level?

[matt grime]

|R| = base_value>1^aleph-0?

what does that mean? base value of what? why isn't 1^aleph-0 1 as it ought to be.
Did you understand why I wrote that 2^alpeh-0=3^alpeh=0 etc based upon the construction of R?


Our stastice is about sets, yours is only about sets of natural numbers

if we have 001001001001... I can tell you the ratio as it's the limit of n/3n as n tends to infinity, ie 1/3.


Why, if I can see that there is a copy of Q inside N, and a copy of N inside Q, can I not conclude they have the same number of elements? Why must the cardinality of a set depend on how it's written down. For instance, the evne integers are 'half the integers' so have card aleph0/2, yet, they are also the set {2n|n in N}, and so they have as many elements as N as well?

[Organic]

There is a conceptual problem in the basis of the "transfinite" world.

By using the word "transfinite" we mean that we can capture the all collection of infinitely many elements where the capturer tool does not belong (transcendent) to the elements which it captures.

R set is a complete set (no gaps) that described as “given any arbitrary interval, this interval includes infinitely many points which are connected to each other”.

Now please show me how infinitely many elements can be both unique AND non-unique (connected) on the same level?

[matt grime]

But you've not correct th problemt that 'capturer tool does not belong to the element it captures'

your aleph0 is still not a natural number.


Don't you in newdigaonlpdf state clearly the rationals are countable, that is |Q|=|N|?

but the even numbers are countable too, so |evens|=|N|, implies aleph0=aleph0/2?

Or would you like to clarify what countable means.

first you state that |R| is 2^aleph-0, then you change that to something that doesn't make sense (just put real numbers in there to see why), and now when asked to clarify that, you come up with something about uniqueness.


What do you mean points of R are connected to each other?
The interval [x,x] contains exactly one point in it.
Q also has these properties, or at least the best guess I can make from what you actually write.

What do you mean by an infinite set whose points are unique and non-unique? where did that come from.

[Organic]

Matt,

'half the integers' so have card aleph0/2, yet, they are also the set {2n|n in N}, and so they have as many elements as N as well?

So aleph0/2 = aleph0 isn't it?

Which means that by your statistics you say: "1".

Also by your statistics 2^aleph0=3^aleph0 --> "1".

Now, if you have these two "1" can you tell me what created each "1"?

[moshek]

I see that it is very exiting here ,
I will join you in few days.

Have fun with mathematics!

Moshek
[:))]

[Organic]

Matt,

What do you mean by an infinite set whose points are unique and non-unique? where did that come from.

As I clearly show here:
http://www.geocities.com/complementarytheory/RiemannsLimits.pdf

No infinitely many unique elements can construct (can be or can use a model of) a solid line.

So you have to deside between infinitely many unique elements XOR solid line.

If you choose infnitely many elements, then their cardinality is the unknown value of infinitely many elements that cannot be completed by definition ("infinite" means no end therefore no completeness).

If you choose solid line then you have no iput that can be used to make math or shortly speaking |R|=?^aleph0.

[Organic]

Matt,

(just put real numbers in there to see why),

base_value is any n>1


aleph0 is the cardinal of any arithmetic series of infinitely many elements.

n>1^aleph0 is the cardinal of any geometirc series of infinitely many elements.

[matt grime]

n>1^aleph0 makes no sense, why is there a "greater than" sign in there, what does it mean?


The real numbers arre equivalence classes of cauchy sequences of rationals, it is a well defined construction.

Take N, take NU{-1}, and NU{-2}

They both have card, in your world, of aleph-0+1, but they are different sets.

The card of {1,2....,n} is the same as {2,3,...,n+1} yet they are different sets. What makes you think cardinality is in anyway a measure of what the elements of the set are?

You claim |N|=|Q| in your cardinal system, and they are different sets too, so what's the point of it?

[Hurkyl]

So, all you have is: there is(=1) or there is not(=0) a bijection between two sets.

That is not all I have; I have other things like well-order types, topologies, algebras, and measures.

But it is frequent that cardinality is what I want. In fact, the idea of there being a map from the natural numbers onto a set is so important that it was given a special name.

[matt grime]

To echo Hurkyl, two sets having the same cardinaltiy is by definition saying there is a bijection between them, that is all it says. You seem to think that different sets cannot ave the same 'cardinality', at least that is how i interpret you assertion that I can't distinguish between aleph-0 and aleph-0+1, well, how can you distinguish between them? what is aleph-0? You assert |Q| and |N| are the same but surely from the way you construct them |Q| = aleph0*log(aleph0) approx.

we have a very good way of distinguishing between N and Q - they are not equal. They are in 1-1 correspondence, though.

[Organic]

Matt,

Please answer to each part of this post.

part 1:


'half the integers' so have card aleph0/2, yet, they are also the set {2n|n in N}, and so they have as many elements as N as well?

So aleph0/2 = aleph0 isn't it?

Which means that by your statistics you say: "1".

Also by your statistics 2^aleph0=3^aleph0 --> "1".

Now, if you have these two "1" can you tell me what created each "1"?


part 2:

n>1^aleph0 makes no sense, why is there a "greater than" sign in there, what does it mean?

By n>1 I mean any n value greater than 1.

The real numbers arre equivalence classes of cauchy sequences of rationals, it is a well defined construction.

It is not well defined construction because it uses simultanuasly two different models that contradict each other (existing on the same level), which are:

1) A model of inifintly many elements.
2) A model of solid line with on gaps.


What makes you think cardinality is in anyway a measure of what the elements of the set are?

dependency Matt,

Can your body exist without the atoms of it?

You claim |N|=|Q| in your cardinal system, and they are different sets too, so what's the point of it?

Again, please read page 5 in:
http://www.geocities.com/complementarytheory/NewDiagonalView.pdf

((aleph0/1)*(1/aleph0))*aleph0=1*aleph0=aleph0


Another important thing:

When I use ", ..." notation in {1, 2, 3, ...} I mean that a set with infinitely many elements cannot be completed, because the meaning of the word "infinite" is "has no end".

[matt grime]

so |R| is in your opinion simultaneously any of n^aleph-0, for any n in N?

Seeing as you are doing arithemic on cardinals, must not |R|=|R|?

thus mustn't 2^aleph-0 = 3^aleph-0 =...?

Otherwise |R| is not equal to |R|.


As for the other questions you raise.

you write that |Q| =|N| trivially because of the counting algorithm you give.

But I can give a counting algorithm that shows |N| is equal to the cardinality of the even natural numbers - the counting goes n <--->2n

Since YOU have said the evens have cardinality aleph0/2, it must then follow if your arithemetic is correct that aleph0 =|N|=|even naturals|=aleph0/2

that is logically what you are claming if you think |Q|=|N|.


And no cardinality does not depend on what the elements in a set are, even in the finite case. I have 4 oranges, I have 4 apples, it's the same 4 in each case.


Who says that the real numbers ARE a line? They are not a line - they are, in one construction, the set of equivalence classes of cauchy sequences of rational numbers. That they are useful for measuring and drawing a line is not important. sqrt -1 is useful in electrical engineering, that doesn't mean it is a voltage or a current.

I've read page 5. It is not correct in showing |Q|=|N| as the correspondence sends both 1/3 and 3 to the same element in N, so it isn't a bijection.

[Organic]

Matt,

Since YOU have said the evens have cardinality aleph0/2, it must then follow if your arithemetic is correct that aleph0 =|N|=|even naturals|=aleph0/2

that is logically what you are claming if you think |Q|=|N|.

((aleph0/1)*(1/aleph0))*aleph0=1*aleph0=aleph0

And no cardinality does not depend on what the elements in a set are, even in the finite case. I have 4 oranges, I have 4 apples, it's the same 4 in each case.

So cardinality value depends on set's content.

|N| is the cardinal of any arithmetic series of infinitely many elements. therefore |Q|=|N|

|R| is the cardinal of any geometric series of infinitely many elements.

|N| or |R| have no unique values, and they are used by me like two common "family names" to infinitely many "first names".

Who says that the real numbers ARE a line?

Please explain to me how to different and unique elements have no gap between them, and the result is NOT a solid line.

'half the integers' so have card aleph0/2, yet, they are also the set {2n|n in N}, and so they have as many elements as N as well?

So aleph0/2 = aleph0 isn't it?

Which means that by your statistics you say: "1".

Also by your statistics 2^aleph0=3^aleph0 --> "1".

Now, if you have these two "1" can you tell me what created each "1"?

[matt grime]

1. I'm going to stop writing aleph-0 because it's just confusing to anyone who might read this. your aleph-0 is not our aleph-0 I will just say A

So let me get this right. A is |N|, now A+1 and A are different? n^A is different for every n in N? |R| is simultaneously 2^A, 3^A, 4^A.... so |R| is not equal to |R|, as it has distinct 'cardinalities' depending on which base you write your decimal expansions in. A is the 'cardinality' of any arithmetic progression, so that A is the 'cardinality' of 2,4,6,8... which is an infinite arithmetic progression, it is also the even numbers and their 'card' is A/2? |Q| is A as |Q| is for some reason the same as A*(1/A)*A

2. Card(S) depends on what the elements are that are in the set too. So that the cardinality of the set of 4 oranges is different from the cardinality of the set of 4 bananas? This is the translation in english of whatever you are thinking in Hebrew.


3. Your 'card' has nothing to do with bijections of sets, and is a many valued function defined on a a set. |R| is many things that you've said are not equal), and have no place in Cantor's Diagonal argument, becuase you have just redefined all the terms in the theorem and its proof, so why should it remain valid?


4. You claim that your 'statistics' distinguish sets, well, N and Q have the same statistics, or at least one in each class of objects |N| and |Q| is the same, thus you don't distinguish between where the cardinalities come from




5. As to the real line. What do you mean by 'different and unique [real numbers] have no element between them'? Given any two distinct real numbers there is always a third between them. Are yo attempting to say there are 'no gaps' in R and it must therefore 'be' a real line? Erm, no, the real numbers are not a line, they are formally equivalence classes of cauchy sequences of real numbers, or Dedekind cuts. They from a totally ordered field that we some times think of as a line, naively, just as the set ZxZ 'is' a lattice point set in the plane.

[Organic]

Matt,

First, thank you very much for this post, it is really a good one.

So let me get this right. A is |N|, now A+1 and A are different? n^A is different for every n in N? |R| is simultaneously 2^A, 3^A, 4^A.... so |R| is not equal to |R|, as it has distinct 'cardinalities' depending on which base you write your decimal expansions in. A is the 'cardinality' of any arithmetic progression, so that A is the 'cardinality' of 2,4,6,8... which is an infinite arithmetic progression, it is also the even numbers and their 'card' is A/2? |Q| is A as |Q| is for some reason the same as A*(1/A)*A

YES.

2. Card(S) depends on what the elements are that are in the set too. So that the cardinality of the set of 4 oranges is different from the cardinality of the set of 4 bananas?

No, cardinals depends on quantity or on arithmethic/geometric progression+operations.

3. Your 'card' has nothing to do with bijections of sets, and is a many valued function defined on a a set. |R| is many things that you've said are not equal), and have no place in Cantor's Diagonal argument, becuase you have just redefined all the terms in the theorem and its proof, so why should it remain valid?

By this model:
http://www.geocities.com/complementarytheory/RiemannsLimits.pdf
I don't see how the transfinite universes can exists between what you call progressions (which I call intersections, in the above model) and
a "solid-line" state of what I call "the strong limit of Math language" (or "actual infinity").

4. You claim that your 'statistics' distinguish sets, well, N and Q have the same statistics, or at least one in each class of objects |N| and |Q| is the same, thus you don't distinguish between where the cardinalities come from

Cardinality is only one parameter, the other parameter is the "structural properties" of |Q| members:
http://www.geocities.com/complementarytheory/UPPs.pdf
that can help us to distinguish between Q and N.

5. As to the real line. What do you mean by 'different and unique [real numbers] have no element between them'? Given any two distinct real numbers there is always a third between them. Are yo attempting to say there are 'no gaps' in R and it must therefore 'be' a real line? Erm, no, the real numbers are not a line, they are formally equivalence classes of cauchy sequences of real numbers, or Dedekind cuts. They from a totally ordered field that we some times think of as a line, naively, just as the set ZxZ 'is' a lattice point set in the plane.

So why standard math uses the word "line" and connect it to a collection of infinitely many elements?

[moshek]

Dear Matt,

I am really exited
from the development
of a real dialog

We only ... try to follow
the real vision of Hilbert
from his famous lecture
at Paris on 1900.

Not the solution of the 23 problems
was his target !
he said that on a very clear way
at the end.

Waiting to join you.

Moshek
[:))]

p.s My paper on 01 laws is almost complete now.

[matt grime]

You have a different definition and a different set of rules relating to cardinals. Therefore, Cantor's argument and its proof, which uses the same symbols but with different meanings, need not remain true.


Example. A function f is unifomly continuous if, for all e greater than zero there is d greater than zero such that |x-y|<d => |f(x)-f(y)| <e.

Now suppose I change the meaning of e 'greater than' 0 so it corresponds in the old terminology to 'e=-1'

then the proof that f(x)=x is uniformly continuous will not remain valid because I've just changed all the meanings around, and infact nothing is uniformly continuous because |a|<-1 is impossible (| | here means modulus, absolute value, not cardinality)

A mathematical quantity is what it does, the aleph-0 of cantorian set theory are an indactor of the isomoprhism type of the set. Two sets have the same cardinality iff they are bijective. This is not true in your meaning of the word cardinality.

you have assigned different meanings and properties to symbols as used by other people to mean radically different things, you cannot expect the translation of every result to hold if you do not translate the meanings.

So you saying the transfinite universe does not exist is the same as saying there are no such things as uniformly continuous functions, when you've redefined all the terms to mean something entirely different.



When elementary mathematics speaks of the number line it is speaking of a mental picture you can draw to learn how to handle the real numbers. It is called a 'line' in higher mathematics because it allows us to do (idealized) geometry, where things behave like the imperfect world of drawing on paper with a pen. We say there are 'no gaps' in the 'line' because by definition the metric space R is complete and therefore every cauchy sequence converges, unlike Q, where, say, the sequence, 3, 3.1, 3.14, 3.141, 3.1415,.... is cauchy but does not converge in Q as pi is not rational.

As such R is a totally well ordered complete field, and is unique (up to order-field-isomorphism) and it can be visualized as a line of elements.

There are other properties we use to distinguish sets in mathematics other than cardinality. Q is not (naturally) well ordered (it can be well ordered with order type w^2 I think), N is.

[Organic]

Matt,

4. You claim that your 'statistics' distinguish sets, well, N and Q have the same statistics, or at least one in each class of objects |N| and |Q| is the same, thus you don't distinguish between where the cardinalities come from

You know what? you a right about |N| and |Q|.
|N| = A
|Q| = Ak/Aj where k or j are independed n's.

[Organic]

Matt,

So, once more why do you think my 01 matrix does not have width aleph0 and length 2^aleph0 from a Cantorian point of view?

For example:


<-------------------Width magnitude =aleph0
{...,3,2,1,0}=Z*
2 2 2 2
^ ^ ^ ^
| | | |
v v v v
{...,1,1,1,1} Length magnitude = 2^aleph0
...,1,1,1,0 |
...,1,1,0,1 |
...,1,1,0,0 |
...,1,0,1,1 |
...,1,0,1,0 |
...,1,0,0,1 |
...,1,0,0,0 |
...,0,1,1,1 |
...,0,1,1,0 |
...,0,1,0,1 |
...,0,1,0,0 |
...,0,0,1,1 |
...,0,0,1,0 |
...,0,0,0,1 |
...,0,0,0,0 |
... V

[matt grime]

When you ask me why i think the number of rows is not 2^aleph-0 in cantorian theory, are you asking me that with MY concept of cardinality? The rows are in bijection with N, since they are formed by a sequence indexed by N, N does not have card 2^aleph-0 in my theory because that statement is equivalent to saying there is a bijection from N to P(N), when no such exists as is easily demonstrated by one of at least 7 proofs that I've seen over the years.

By definition two sets have the same cardinality IFF they are isomorphic as sets.

[Organic]

Matt,

It is very nice what you say, but now there is no ZF axiom of infinity in my new construction, so the matrix in my previous post must have a length magnitude of 2^aleph0, isn't it?

[matt grime]

Damn it iI just wiped my reply.

Ok


If you are stating that the number of rows has card 2^aleph-0 and that you are using these terms in the conventional way that mathematics DEFINES them, then you are stating that there is a bijection between N and its power set. Now you know there isn't one in the case of finite sets, and the same proof states the same in the infinite case if written properly, as does the observation that the reals do not have measure zero.

However, we have established that you are not using aleph-0 in its conventional sense, so in your theory who knows what happens. The important thing is to realize that the things you are talking about are not the things a mathematician talks about.

It is by definition that two sets have the same cardinality iff they are bijective, and by definition that 2^aleph-0 is the cardinality of the power set of N. We could have declared aleph-1 to be the cardinality of the power set of N, but we didn't because the statement that the cardinality of the power set of N is the 'smallest' uncountable cardinal is independent of ZF! (Cohen et al, the continuum hpothesis). There is no bijection between N and P(N) therefore we DECLARE them to have different cardinalities.


Cardinality is not some abstract concept independent of alephs that we 'model' with alephs, they are inextricably bound.


So it is because you refuse to accept a definition that you are apparently contradicting mathematics. You cannot contradict a definition, only state that it does not do what you want.

You should then offer a different label for a different object. Your aleph-0 is not the aleph-0 of mathematics, it does not behave the same way and does not encode the same information, which is simply the isomorphism class of the set

[Organic]

Matt,

In this post aleph0 is the Cantorian aleph0, so in this case langth magnitude cannot be but 2^aleph0, as we clearly can see here:

<-------------------Width magnitude = aleph0
{...,3,2,1,0}=Z*
2 2 2 2
^ ^ ^ ^
| | | |
v v v v
{...,1,1,1,1} Length magnitude = 2^aleph0
...,1,1,1,0 |
...,1,1,0,1 |
...,1,1,0,0 |
...,1,0,1,1 |
...,1,0,1,0 |
...,1,0,0,1 |
...,1,0,0,0 |
...,0,1,1,1 |
...,0,1,1,0 |
...,0,1,0,1 |
...,0,1,0,0 |
...,0,0,1,1 |
...,0,0,1,0 |
...,0,0,0,1 |
...,0,0,0,0 |
... V


Please tell me what do you thing?

[matt grime]

So you are saying that you believe there is a bijection between P(N) and N? Which is what you mean if you are saying there is a bijection between a set of rows that are in 1-1 correspondence with N If you are using aleph-0 properly. This is wrong. It is easily and variously proven to be wrong. As any person with the most basic understanding of mathematical convetion knows.

[Organic]

No Dear Matt,

This is the beautiful thing in Math, you don't have to believe in anything and nothing is wrong or right.

All we have is consistent(=interesting) system or non-consistent(=non-interesting) system, no less no more.

As you can see by the list below, there are infinitely many information structures which are beyond the scope of N members, but the structural arrangement of them give us the possibility to construct a list of infinitely many unique sequences.

The length of this list has a magnitude of 2^aleph0(the Cantorian one)
and as we can clearly see, it is enumerable.


<----------------------Width magnitude = aleph0
{...,3,2,1,0} = Z*
2 2 2 2
^ ^ ^ ^
| | | |
v v v v
{...,1,1,1,1} <--> 1 Length magnitude = 2^aleph0
...,1,1,1, |
...,1,1, , |
...,1,1, , |
...,1, , , |
...,1, , , |
...,1, , , |
...,1, , , |
... |
...,0,0,0,0 <--> 2 |
...,0,0,0, |
...,0,0, , |
...,0,0, , |
...,0, , , |
...,0, , , |
...,0, , , |
...,0, , , |
... |
...,1,1,1,1 <--> 3 |
...,1,1,1, |
...,1,1, , |
...,1,1, , |
...,1, , , |
...,1, , , |
...,1, , , |
...,1, , , |
... |
...,0,0,0,0 <--> 4 |
...,0,0,0, |
...,0,0, , |
...,0,0, , |
...,0, , , |
...,0, , , |
...,0, , , |
...,0, , , |
... |
...,1,1,1,1 <--> 5 |
...,1,1,1, |
...,1,1, , |
...,1,1, , |
...,1, , , |
...,1, , , |
...,1, , , |
...,1, , , |
|
... V


Let us see what is the connaction between a Binary list and the above list.

First let us look at this list:

<---arithmetic magnitude
{...,3,2,1,0} = Z*
2 2 2 2
^ ^ ^ ^
| | | |
v v v v
{... 1 1 1 1} geometric magnitude
... 1 1 1 |
... 1 1 1 |
... 1 1 / |
... 1 1 1 |
... 1 1 / |
... 1 //1 |
... 1 // / |
... 1 1|1 |
... 1 1| |
... 1 ||1 |
... 1 // |
... /|1 1 |
... / |1 |
... | || 1 |
... | || |
... 1 || V

the same can be done with '0' notations

Shotly speaking the main structure here is:

<---arithmetic magnitude
{...,3,2,1,0} = Z*
2 2 2 2
^ ^ ^ ^
| | | |
v v v v
{... 1 1 1 1} geometric magnitude
... 1 1 1 |
... 1 1 |
... 1 1 |
... 1 |
... 1 |
... 1 |
... 1 |
... |
... |
... |
... |
... |
... |
... |
... |
... V

again, the same can be done with '0' notations

Now we shall show thet this information is greather then aleph0 magnitude.

Step 1: we will show again our list in this way:

<---arithmetic magnitude
{...,3,2,1,0} = Z*
2 2 2 2
^ ^ ^ ^
| | | |
v v v v
{...,1,1,1,1} geometric magnitude
...,1,1,1,0 |
...,1,1,0, |
...,1,1,0, |
...,1,0, , |
...,1,0, , |
...,1,0, , |
...,1,0, , |
...,0, , , |
...,0, , , |
...,0, , , |
...,0, , , |
...,0, , , |
...,0, , , |
...,0, , , |
...,0, , , |
... V

step 2: to make it clearer we shall show it now in this way:

<---arithmetic magnitude
{...,3,2,1,0} = Z*
2 2 2 2
^ ^ ^ ^
| | | |
v v v v
{...,1,1,1,1} <--> 1 geometric magnitude
...,1,1,1, |
...,1,1, , |
...,1,1, , |
...,1, , , |
...,1, , , |
...,1, , , |
...,1, , , |
... |
...,0,0,0,0 <--> 2 |
...,0,0,0, |
...,0,0, , |
...,0,0, , |
...,0, , , |
...,0, , , |
...,0, , , |
...,0, , , |
... |
...,1,1,1,1 <--> 3 |
...,1,1,1, |
...,1,1, , |
...,1,1, , |
...,1, , , |
...,1, , , |
...,1, , , |
...,1, , , |
... |
...,0,0,0,0 <--> 4 |
...,0,0,0, |
...,0,0, , |
...,0,0, , |
...,0, , , |
...,0, , , |
...,0, , , |
...,0, , , |
... |
...,1,1,1,1 <--> 5 |
...,1,1,1, |
...,1,1, , |
...,1,1, , |
...,1, , , |
...,1, , , |
...,1, , , |
...,1, , , |
|
... V

[matt grime]

I'm not entirely sure what the picture you draw now is meant to be, or what's going on with it, but that isn't really important.

consistent doesn't mean interesting.

your use of alephs is inconsistent (and thus uninteresting...?) with the correct mathematical usage. And there are conventions that we must keep to.

you have just stated that there is a bijection from N to P(N); that the whole of measure theory is wrong; that Baire's category theory needs rethinking; that the principle of least upper bound makes no sense.

[Organic]

Matt,

Please forget about my point of view on aleph0.

In my last post and in this post I am talking about the standard Cantorian meaning of aleph0.

I made a pdf of my last post, maybe you will find it clearer:

http://www.geocities.com/complementarytheory/Countable.pdf


you have just stated that there is a bijection from N to P(N); that the whole of measure theory is wrong; that Baire's category theory needs rethinking; that the principle of least upper bound makes no sense.

I know that for more than 20 years.

[matt grime]

Then you have been mistaken about mathematics for more than 20 years.

[Hurkyl]

Can you tell us what your construction is, instead of showing a tiny piece of it?

[Organic]

Matt,

Prove that my matrix does not have a length with 2^aleph0 magnitude.

[Organic]

Hi Hurkyl,

Please look at this pdf:
http://www.geocities.com/complementarytheory/Countable.pdf

read it carefully and when you undestand it please try to address it in standard Math notations.

Thank you,

Organic

[Hurkyl]

I repeat, can you tell us what your construction is, rather than showing a small corner of the array and assuming the rest is obvious?

[matt grime]

Originally posted by Organic
Matt,

Prove that my matrix does not have a length with 2^aleph0 magnitude.


Suppose that there is a set, R of cardinality 2^aleph-0, and that there is a function from N to R.

We may replace R with P(N) as by definition R is in bijective correspondence with it, and thus we have a map f from N to P(N).

Define T in P(N) by t in T iff t not in f(t)

T is by the usual argument not in ran(f), hence there is no suejective set map from N to P(N). Thus R is not countable.

[Organic]

Matt,

Please don't repeat again on your MANTRA.

You can prove that my list does not have a length of magnitude 2^aleph0 iff you can prove thet Z* /= Z* .

[matt grime]

Why mantra? It is a proof.

Two sets have the same cardinality IFF they are in bijective correspondence, that is the definition of what it means for two sets to have the same cardinality. 2^aleph-0 is, by definition, the cardinality of the power set of N. Cardinality is purely a statement about isomorphisms, not what ever you have in mind. Thus you are stating there is a bijection from N to P(N) if you are stating that the set of rows simultaneously has cardinality aleph-0 (is in bijection with N) AND 2^aleph-0 (is in bijection with P(N). If there is a bijection from A to B and a bijection from A to C there is a bijection from B to C.)

You have to play by my rules on this one; you said so yourself.

[Organic]

Matt,

|Z*|<|P(Z*)| but both are contable.

[matt grime]

No, there is no bijection from N to P(Z*), which I believe you define to be {0,1,2...}.

An (infinite) set is COUNTABLE is, by _definition_, stating that the is a bijection from N to that set. There is no bijection from any set to its power set.

You do know what countable means?

[Organic]

Matt,

You are playing with words (definitions) I show a concrete proof
that |Z*|<|P(Z*)| and both ( Z* and P(Z*) ) are enumerable.

[matt grime]

You cannot do that if you are using the definitiions correctly.

All your constructions start with some finite portion of a picture, and then claim that in the ellipsis everything hangs together.

How have you shown P(N) is countable? it was by those diagrams in newdiagonal.pdf, the ones that I proved had exactly enumerated the finite subsets of N.

Or have you got another 'proof' that there is a bijection from N to P(N)

write it here then, and let's start the same old tired arguments all over again...

[Organic]

Matt,

You still don't understand do you?

To contradict my last version, that can be found here:

http://www.geocities.com/complementarytheory/Countable.pdf

YOU HAVE TO PROVE THAT Z* IS NOT EQUAL TO Z* !!!


kappish?

[matt grime]

You give no reason as to why the rows form a set of cardinality 2^aleph-0 - that is that they are in bijection with P(N), indeed you aren't even saying what the rows are, and how you have constructed them. You merely state that they must be a set of cardinality 2^aleph-0. As far as I can see there *might* be a countable set of elements for each n in N, and that you are taking thier union. A countable union of countable sets is countable.

Who knows what horrors your ellipsis hides. Apparently not even you because you cannot/do not explain it.


Your picture is very unclear - where is the bijection with N? I see some rows are labelled with elements of N, most aren't, what are these rows, how are you constructing them, why are they countable - give the injection to N, why are they of cardinality 2^aleph-0 or greater? give an injection from P(N).

Now, if your constructions, whatever they are are true (they aren't) then my proof that there is no bijection from N to P(N) is incorrect. where is it wrong? Where do all the proofs of this fact go wrong? Mine is a clear simple proof well known and easily checked with all its terms defined. Your pictures are what? they are fragments of something larger that you refuse to explain how to construct, and its properties are not verified, merely stated.



here, let me do this, consider the set {1,2,4,67,84....} this is clearly a countable set of real numbers and contains a brown fox. prove me wrong.

[Organic]

Matt,

You protect yourself from being hurt by the consequences of my proof that R is enumerable.

So please don’t ask me because for me what I did is clear as a middle noon sun.

Please print my proof, and ask your colleges about it.

[suyver]

Originally posted by Organic
Matt,

You protect yourself from being hurt by the consequences of my proof that R is enumerable.

So please don’t ask me because for me what I did is clear as a middle noon sun.

Please print my proof, and ask your colleges about it.

At what point is this considered trolling?

[Organic]

Dear Matt,


Please look again at my paper, I put more details that can make
it clearer:

http://www.geocities.com/complementarytheory/Countable.pdf


Yours,

Organic

[matt grime]

Originally posted by suyver
At what point is this considered trolling?

can cranks be trolls? for a while he was in danger of making sense (the sense being 'I've completely rewritten all the rules and look at the problems that causes!'), though the last few posts have gone back to being just irrational assertions and refusal to acknowledge that defintions are somehow, well, erm, defined.

The average response to organic's 'proof' (ie a picture with a couple of labels) by my 'colleges' (one of the best typos I've seen) would be 'idiot' probably preceded by one of several old fashioned expletives. I myself favour a simple, muttered, 'tit' when confronted with these things in the best Peter Kay as Brian Potter way. (Apologies to everyone outside of England/UK who has no idea who that is. try getting hold of Phoenix Nights it's not half bad).

The odd thing is his diagrams keep getting more and more elaborate, each time it is easily pointed out where he's going wrong; yet he never addresses these problems, just comes back with yet another one with more things going on and even fewer explanations. And the latest one always proves mathematics wrong! Ignore the proof that it's rubbish, and that each previous one was given just the same fanfare on arrival and was just as easily dismissed as the ravings of an idiot. If he is a troll he's got a lot of free time to devise these things. I'm a professional mathematician, this is a distraction from research and each rebuttal takes about a minute to devise; he must spend hours coming up with these pretty pdfs.

[Organic]

Ok Matt,

After you air your view, please look again at page 3 (step 1):

http://www.geocities.com/complementarytheory/Countable.pdf

[matt grime]

Nope, there's still no reason at all to conclude that the list is in anyway a set of card 2^aleph-0, there is no reason to suppose it contains all the elements of the set of all infinite strings of 0s and 1s, indeed the list STILL contains only those strings that have a finite number of 0s on them as has been proven to you. Your only proof is that it can be no other thing... erm, not true. As I've asked, and Hurkyl, where is the string ..1010101 of alternating 0s and 1s?

So, I've read the article AGAIN. WHy don't you explain where the counter proofs of you assertions are wrong in your opinion. Remember when we asked how to construct the diagram? And we agreed the th first column is (1010101010... ) the second (110011001100..) and so on - the nth is 2^n ones, 2^n 0s, looping again and again?

remember how we showed you that that implies that every row has only a finite number of zeroes in it? remember how that implies the string ...01010101) with an infinite number of 0s in it is not on the list? remember? come on, we;ve read the article, we've said what we consider wrong with it, and it is encapsulated in this paragraph and the previous one. so where are we wrong. come on, explain it in clear simple words for us that can't understand your maths, tell us where we 've gone wrong in the analysis of the diagram.

[Organic]

Matt,

...01010101 or ...10101010 is in the list, for example:

Let us take again our set:

{...,3,2,1,0}=Z*
2 2 2 2
^ ^ ^ ^
| | | |
v v v v
{...,1,1,1,1}<--> 1
...,1,1,1,0 <--> 2
...,1,1,0,1 <--> 3
...,1,1,0,0 <--> 4
...,1,0,1,1 <--> 5
...,1,0,1,0 <--> 6
...,1,0,0,1 <--> 7
...,1,0,0,0 <--> 8
...,0,1,1,1 <--> 9
...,0,1,1,0 <--> 10
...,0,1,0,1 <--> 11
...,0,1,0,0 <--> 12
...,0,0,1,1 <--> 13
...,0,0,1,0 <--> 14
...,0,0,0,1 <--> 15
...,0,0,0,0 <--> 16
...

Now let us make a little redundancy diet:

{...,3,2,1,0}=Z*
2 2 2 2
^ ^ ^ ^
| | | |
v v v v
... 1-1-1-1 <--> 1
\ \ \0 <--> 2
\ 0-1 <--> 3
\ \0 <--> 4
0-1-1 <--> 5
\ \0 <--> 6
0-1 <--> 7
\0 <--> 8
... 0-1-1-1 <--> 9
\ \ \0 <--> 10
\ 0-1 <--> 11
\ \0 <--> 12
0-1-1 <--> 13
\ \0 <--> 14
0-1 <--> 15
\0 <--> 16
...

and we get:

{...,3,2,1,0}=Z*
2 2 2 2
^ ^ ^ ^
| | | |
v v v v
/1 <--> 1
1
/ \0 <--> 2
1
/\ /1 <--> 3
/ 0
/ \0 <--> 4
... 1
\ /1 <--> 5
\ 1
\/ \0 <--> 6
0
\ /1 <--> 7
0
\0 <--> 8

/1 <--> 9
1
/ \0 <--> 10
1
/\ /1 <--> 11
/ 0
/ \0 <--> 12
... 0
\ /1 <--> 13
\ 1
\/ \0 <--> 14
0
\ /1 <--> 15
0
\0 <--> 16
...

[matt grime]

That isn't one of the rows though is it? It is the set of rows that needs to have cardinality both aleph-0 and 2^aleph-0, not the number of ways of choosing entries from the rows, or the number of paths through the rows.


Please answer this question.

Is the description of the construction of the diagram I gave accurate? The one I gave two posts back, the one I've given several times.

YES or NO? Can't say fairer than that; all we want in your next post is exactly on word, yes, or no, which is it?

[Organic]

Matt,

That isn't one of the rows though is it? It is the set of rows that needs to have cardinality both aleph-0 and 2^aleph-0, not the number of ways of choosing entries from the rows, or the number of paths through the rows.

My (aleph0 x 2^aleph0) matrix and an Infinitely (Width x Length) Binary Tree are two representations of the same thing.

[matt grime]

Is the description of the construction of the diagram I gave accurate? The one I gave FOUR posts back, the one I've given several times.

YES or NO? Can't say fairer than that; all we want in your next post is exactly on word, yes, or no, which is it?

[Organic]

Matt,

1) And we agreed the th first column is (1010101010... ) the second (110011001100..) and so on - the nth is 2^n ones, 2^n 0s, looping again and again?

2) remember how we showed you that that implies that every row has only a finite number of zeroes in it? remember how that implies the string ...01010101) with an infinite number of 0s in it is not on the list?


1) Yes I agree that this is the redundant way to show how the columns are constructed, but:

2) You showed nothing because the three representations below are one and only one thing:

{...,3,2,1,0}=Z*
2 2 2 2
^ ^ ^ ^
| | | |
v v v v
{...,1,1,1,1}<--> 1
...,1,1,1,0 <--> 2
...,1,1,0,1 <--> 3
...,1,1,0,0 <--> 4
...,1,0,1,1 <--> 5
...,1,0,1,0 <--> 6
...,1,0,0,1 <--> 7
...,1,0,0,0 <--> 8
...,0,1,1,1 <--> 9
...,0,1,1,0 <--> 10
...,0,1,0,1 <--> 11
...,0,1,0,0 <--> 12
...,0,0,1,1 <--> 13
...,0,0,1,0 <--> 14
...,0,0,0,1 <--> 15
...,0,0,0,0 <--> 16
...

Now let us make a little redundancy diet:

{...,3,2,1,0}=Z*
2 2 2 2
^ ^ ^ ^
| | | |
v v v v
... 1-1-1-1 <--> 1
\ \ \0 <--> 2
\ 0-1 <--> 3
\ \0 <--> 4
0-1-1 <--> 5
\ \0 <--> 6
0-1 <--> 7
\0 <--> 8
... 0-1-1-1 <--> 9
\ \ \0 <--> 10
\ 0-1 <--> 11
\ \0 <--> 12
0-1-1 <--> 13
\ \0 <--> 14
0-1 <--> 15
\0 <--> 16
...

and we get:

{...,3,2,1,0}=Z*
2 2 2 2
^ ^ ^ ^
| | | |
v v v v
/1 <--> 1
1
/ \0 <--> 2
1
/\ /1 <--> 3
/ 0
/ \0 <--> 4
... 1
\ /1 <--> 5
\ 1
\/ \0 <--> 6
0
\ /1 <--> 7
0
\0 <--> 8

/1 <--> 9
1
/ \0 <--> 10
1
/\ /1 <--> 11
/ 0
/ \0 <--> 12
... 0
\ /1 <--> 13
\ 1
\/ \0 <--> 14
0
\ /1 <--> 15
0
\0 <--> 16
...

[matt grime]

So we have absolutely nailed the construction of the array.

Now, are you claiming the the cardinality of the rows is 2^aleph-0 because for every element in the power set of N, there is a row which corresponds to the indicator function of that element in the power set? Yes, or No is again all that is required. Note this is equivlaent to saying that the set of rows has card 2^aleph-0 because the rows form a 'complete' list of every string of 0s and 1s, ie *ANY* possible string of 0s and 1s occurs as one of the rows.

[Organic]

Matt,

If we use your terminology, then the answer is yes.

[matt grime]

But we agreed that we are using the cantor view point on cardinality, didn't we?

So, let z be some arbitrary element in P(N), it must if your conjecture is true be one of the rows. Let the number of the row (they're enumerated by N) be R.

Are we ok so far? Yes or no?

[Organic]

Matt,

(they're enumerated by N)

No, they're enumerated.

This is what you don't understand, the list of unique notations that related to R is longer then the list of unique notations that related to N.

In your termenology the complete R list is longer than the complete N list.

[matt grime]

So the rows are not of cardinality aleph-0 as you've been claiming all through this, then?

[Organic]

Matt,

You simply don't read what I write.

So here it is again:

1) Both R list and N list are enumerable.

2) |R|>|N|

[matt grime]

So, when I asked you if the rows could be enumerated by N, and you said no, were you lying or mistaken? because that's what enumerable means in this context. so we ask again, are the set of rows in bijective correspondence with N as you claim they are? So picking an element z in P(N) it occurs at row r for some r in N (this r was the R last time, i didn't mean R as in real number, sorry if that's the confusion), that is what you mean by the rows are enumerable, which they are by the construction weve agreed on. And you are claiming that each element of P(N) corresponds to some row.

[matt grime]

Incidentally, the word enumerable is the same as countable. So, remembering that we are usgin real mathematics here, your last post states there is a bijection between R and N, and that there isn't a bijection between R and N.

[Organic]

Matt,

are the set of rows in bijective correspondence with N as you claim they are?

You know what? let us play your game.

1) I constructed a list with unique sequences of 0 1 combinations each, which its length has 2^aleph0 magnitude.

2) I also showed a bijection from N to P(N).


Conclusion:

The transfinite univereses do not exist.

[matt grime]

But we are playing my game, we agreed to use the proper definitions of all the words.

So there is a bijectin from N to P(N) because there is a row in your diagram for each element of the power set - the element being given by the indicator function described by each row. that is the bijection isn't it?

[Organic]

Matt,

We have found that |P(N)|>=|N|

Therefore the tansfinite cardinality does not exist.

[matt grime]

Please answer my question as to how the bijection you are claiming between N and P(N) arises.

here it is


So [in your opinion] there is a bijection from N to P(N) because there is a row in your diagram for each element of the power set - the element being given by the indicator function described by each row. that is the bijection isn't it?

again all that is required is a simple yes or no answer, no more.

[Organic]

Matt,

There is no YES/NO answer here.

Conclusions:


1) |R| > |N| (R list is longer then N list).

2) There is no necessary connection between N and being enumerable.

3) The words “complete” or “all” are meaningless when they are related to a collection of infinitely many elements, because they are leading to contradiction (|R| >= |N|).

Instead, the word "any is used.

Maybe this will help:


<--------------------Arithmetic magnitude

{...,3,2,1,0} = Z*
2 2 2 2
^ ^ ^ ^ Geometric magnitude
| | | | |
v v v v |
{...,1,1,1,1} <--> 1 |
...,1,1,1, |
...,1,1 <---- Cantor’s |
...,1,1, , Diagonal |
...,1, , , |
...,1, , , |
...,1, , , |
...,1, , , |
... |
...,0,0,0,0 <--> 2 |
...,0,0,0, ----. |
...,0,0, , | |
...,0,0, , | Not Covered |
...,0, , , |-- By N |
...,0, , , | |
...,0, , , | |
...,0, , , | |
... -----------‘ |
...,1,1,1,1 <--> 3 |
...,1,1,1, |
...,1,1, , |
...,1,1, , |
...,1, , , |
...,1, , , |
...,1, , , |
...,1, , , |
... |
...,0,0,0,0 <--> 4 |
...,0,0,0, |
...,0,0, , |
...,0,0, , |
...,0, , , |
...,0, , , |
...,0, , , |
...,0, , , |
... |
...,1,1,1,1 <--> 5 |
...,1,1,1, |
...,1,1, , |
...,1,1, , |
...,1, , , |
...,1, , , |
...,1, , , |
...,1, , , |
|
... V

[matt grime]

I would like to remind you that you are operating in the world of conventional mathematics and therefore all of the reasons you just gave are irrelevant, not to say misleading and erroneous.

You have stated there is a bijection between the set of rows and P(N), yet you cannot give this bijection or offer an existence proof. As your claims are supposedly constructive this is obviously a big error on your part. I can predict you will accuse me of just playing with words and misunderstanding the 'true' picture of what maths is, but I think we've seen enough of your position to see it is indefensible; you certainly haven't been able to defend it.

In particular enumerable and countable are defined by bijections with the set of Natural numbers, irrespective of what your opinion is on the matter.

Just because one set appears 'bigger than another' is not important. You think that |Q| equals |N| in whatever meaning you give to those symbols, yet clearly one is 'smaller' than the other, in the sense that a list of Q elements contains a list of N elements as a sublist, to put it in your terms.

After all this do you think that you perhaps ought to learn the meaning of any of the terms you use, because you evidently haven't even after somewhere in the region of 600 posts on it to my knowledge.

[Organic]

Matt,

There exist a countable list of all(=your termenology) R members represented by infinitely long combinations of 01 sequences, in front of your eyes.

Prove that this list does not exist, but before that take a look again on:

<--------------------Arithmetic magnitude

{...,3,2,1,0} = Z*
2 2 2 2
^ ^ ^ ^ Geometric magnitude
| | | | |
v v v v |
{...,1,1,1,1} <--> 1 |
...,1,1,1, |
...,1,1 <---- Cantor’s |
...,1,1, , Diagonal |
...,1, , , |
...,1, , , |
...,1, , , |
...,1, , , |
... |
...,0,0,0,0 <--> 2 |
...,0,0,0, ----. |
...,0,0, , | |
...,0,0, , | Not Covered |
...,0, , , |-- By N |
...,0, , , | |
...,0, , , | |
...,0, , , | |
... -----------‘ |
...,1,1,1,1 <--> 3 |
...,1,1,1, |
...,1,1, , |
...,1,1, , |
...,1, , , |
...,1, , , |
...,1, , , |
...,1, , , |
... |
...,0,0,0,0 <--> 4 |
...,0,0,0, |
...,0,0, , |
...,0,0, , |
...,0, , , |
...,0, , , |
...,0, , , |
...,0, , , |
... |
...,1,1,1,1 <--> 5 |
...,1,1,1, |
...,1,1, , |
...,1,1, , |
...,1, , , |
...,1, , , |
...,1, , , |
...,1, , , |
|
... V

As you can see, Cantor's diagonal does not cover this list and also
N members do not cover it, therefore there is no bijection
from |N| to |P(N)| (|P(N)|>|N|).

Therefore the "bijection" that you see between R and N is only an illusion, because each
n <--> ...000/...111 capture only one slice form each 0 or 1 bulk of geometric magnitude, as you can clearly see below:

<---Arithmetic magnitude

{...,3,2,1,0} = Z*
2 2 2 2
^ ^ ^ ^
| | | |
v v v v
{...,1-1-1-1} <--> 1 Geometric magnitude(based on the
...,1,1,1,0 <--> 2 | thin notations)
...,1,1,0/ |
...,1,1/0, |
...,1,0, , |
...,1/0, , |
...,1|0, , |
...,1|0, , |
...,0/ , , |
...,0, , , |
...,0, , , |
...,0, , , |
...,0, , , |
...,0, , , |
...,0, , , |
...,0, , , |
... V


As for Q and N, both have an arithmetic magnitude.

[matt grime]

So now you admit there is no bijection from N to P(N)? So why did you claim that aleph-0 equals 2^aleph-0? Have you completely recanted your assertion, or are you going to claim that you've never insisted that they are the same? becaue you've repeatedly insisted that the reals are countable that implies there is a bijection from N to P(N).

Or are you going to simultaneously claim that there is a set of cardinality 2^aleph-0 that is countable, yet not in bijection with N, contradicting the definition of countable?

[Organic]

Matt,


I said that there is a bijection from |N| to |P(N)| when I play your game, and by your game the conclusion is that the transfinite cardinality cannot exist because we get |N|=<|P(N)|.

But if we want to understand what is going on, we have no choice but to say bye bye to your game and move to another game, where R is enumerable but |R|>|N|.

This result, which is impossible trough your game, can be simply and clearly explained and demonstrated by my game.


Now we are in my game, so this time please read all that is below and try to understand it:

There exist a countable list of all(=your terminology) R members represented by infinitely long combinations of 01 sequences, in front of your eyes.

Prove that this list does not exist, but before that take a look again on:

<--------------------Arithmetic magnitude

{...,3,2,1,0} = Z*
2 2 2 2
^ ^ ^ ^ Geometric magnitude
| | | | |
v v v v |
{...,1,1,1,1} <--> 1 |
...,1,1,1, |
...,1,1 <---- Cantor’s |
...,1,1, , Diagonal |
...,1, , , |
...,1, , , |
...,1, , , |
...,1, , , |
... |
...,0,0,0,0 <--> 2 |
...,0,0,0, ----. |
...,0,0, , | |
...,0,0, , | Not Covered |
...,0, , , |-- By N |
...,0, , , | |
...,0, , , | |
...,0, , , | |
... -----------‘ |
...,1,1,1,1 <--> 3 |
...,1,1,1, |
...,1,1, , |
...,1,1, , |
...,1, , , |
...,1, , , |
...,1, , , |
...,1, , , |
... |
...,0,0,0,0 <--> 4 |
...,0,0,0, |
...,0,0, , |
...,0,0, , |
...,0, , , |
...,0, , , |
...,0, , , |
...,0, , , |
... |
...,1,1,1,1 <--> 5 |
...,1,1,1, |
...,1,1, , |
...,1,1, , |
...,1, , , |
...,1, , , |
...,1, , , |
...,1, , , |
|
... V

As you can see, Cantor's diagonal does not cover this list and also
N members do not cover it, therefore there is no bijection
from |N| to |P(N)| (|P(N)|>|N|).

Therefore the "bijection" that you see between R and N is only an illusion, because each
n <--> ...000/...111 capture only one slice form each 0 or 1 bulk of geometric magnitude, as you can clearly see below:

<---Arithmetic magnitude

{...,3,2,1,0} = Z*
2 2 2 2
^ ^ ^ ^
| | | |
v v v v
{...,1-1-1-1} <--> 1 Geometric magnitude(based on the
...,1,1,1,0 <--> 2 | thin notations)
...,1,1,0/ |
...,1,1/0, |
...,1,0, , |
...,1/0, , |
...,1|0, , |
...,1|0, , |
...,0/ , , |
...,0, , , |
...,0, , , |
...,0, , , |
...,0, , , |
...,0, , , |
...,0, , , |
...,0, , , |
... V


As for Q and N, both have an arithmetic magnitude.



Conclusions:


1) |R| > |N| (R list is longer then N list).

2) There is no necessary connection between N and being enumerable.

3) The words “complete” or “all” are meaningless when they are related to a collection of infinitely many elements, because they are leading to contradiction (|R| >= |N|).

Instead, the word "any" is used.

[matt grime]

I am perfectly open to new ideas, always have been, however, you have repeatedly said that proper mathematics is wrong and its error means that Cantor is wrong, and that there is only one kind of infinity. It doesn't matter what your definitions and issues are here, only that you must then prove something is wrong in Cantor's argument. Your method is to then redefine all of the terms to mean something else. Thus you are being a complete moron in claiming he is wrong, because you are deliberately misinterpreting what he said.


I don't particularly care for your opinion on your new mathematics, I don't see anything interesting in it, nor new. My only position has been that you do not understand tht which you defame and that we cane easliy prove Cantor is correct because you evidently don't have a clue what's going on. Shall we agree on that? I don't care about your opinion, and you know squat about mathematics?

There is no bijection in my mathematical world from N to P(N). So why do you keep saying that? It's always been your contention that there must be, but you've never been able to prove it, and now you're claiming you never wanted to...? Pillock.

[Organic]

Matt,

Ok, prove by your system that my matrix does not have the complete 01 combinations.

...0101 or ...1010 is in the list, for example:

Let us take again our set:

{...,3,2,1,0}=Z*
2 2 2 2
^ ^ ^ ^
| | | |
v v v v
{...,1,1,1,1}<--> 1
...,1,1,1,0 <--> 2
...,1,1,0,1 <--> 3
...,1,1,0,0 <--> 4
...,1,0,1,1 <--> 5
...,1,0,1,0 <--> 6
...,1,0,0,1 <--> 7
...,1,0,0,0 <--> 8
...,0,1,1,1 <--> 9
...,0,1,1,0 <--> 10
...,0,1,0,1 <--> 11
...,0,1,0,0 <--> 12
...,0,0,1,1 <--> 13
...,0,0,1,0 <--> 14
...,0,0,0,1 <--> 15
...,0,0,0,0 <--> 16
...

Now let us make a little redundancy diet:

{...,3,2,1,0}=Z*
2 2 2 2
^ ^ ^ ^
| | | |
v v v v
... 1-1-1-1 <--> 1
\ \ \0 <--> 2
\ 0-1 <--> 3
\ \0 <--> 4
0-1-1 <--> 5
\ \0 <--> 6
0-1 <--> 7
\0 <--> 8
... 0-1-1-1 <--> 9
\ \ \0 <--> 10
\ 0-1 <--> 11
\ \0 <--> 12
0-1-1 <--> 13
\ \0 <--> 14
0-1 <--> 15
\0 <--> 16
...

and we get:

{...,3,2,1,0}=Z*
2 2 2 2
^ ^ ^ ^
| | | |
v v v v
/1 <--> 1
1
/ \0 <--> 2
1
/\ /1 <--> 3
/ 0
/ \0 <--> 4
... 1
\ /1 <--> 5
\ 1
\/ \0 <--> 6
0
\ /1 <--> 7
0
\0 <--> 8

/1 <--> 9
1
/ \0 <--> 10
1
/\ /1 <--> 11
/ 0
/ \0 <--> 12
... 0
\ /1 <--> 13
\ 1
\/ \0 <--> 14
0
\ /1 <--> 15
0
\0 <--> 16
...


you have repeatedly said that proper mathematics is wrong and its error means that Cantor is wrong, and that there is only one kind of infinity.

My system is reacher then Cantor's transfinite universes bacause:

1) By my system aleph0+1 > aleph0 , 2^aleph0 < 3^aleph0

2) By Cantor's system aleph0+1 = aleph0 , 2^aleph0 = 3^aleph0


By the way, when we move from the 01 matrix representation to the Binary Tree representation, the meaning of the word magnitude become clearer, because several sequential 1 or 0 notations of each column in the matrix, are compressed to a single notation, which its magnitude equivalent to the quantity of the notations that it represents.

[moshek]

My new motto will be don't feed the trolls.


sory i did not answer you matt
i am to busy

but way you call Organic a Troll ?

Don't you see he is just a monkey ...

And aren't we all the monkey of Euclid ?


Moshek
[:))]