What is the Collatz Problem and how can it be solved?

[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 shouldn't 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 infinite 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, which 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.