How Does Complementary Logic Redefine Mathematical Infinity?

[matt grime]

Originally posted by Organic
You play with the words I use instead of try to understand their meaning.

This is what I call scholastic approach.


By mutation I mean that there is a change in some already used thing that deeply change its proprties.

Teach by example:

let S by the subset of Natural numbers that are uninteresting. S has a least element (not all numbers are interesting), yet this least element is now interesting as it is the least interesting positive number.

There are many such paradoxes lying around all of the to do with imprecise definitions.

But not only do you use imprecise definitions but fail to use correctly the precise ones that exist.

Clear up the mistakes in New Diagonal before committing any more

 

[Organic]

You are talking to yourself, not to me because of a very simple reason.

For me the meaning of "what is a number?" starts here:

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

For you numbers are objects that do not depends on your ability
to define them.

For you mathematical objects are actual objects, based on unchanged
logic terms.

For me Mathematics is only a form of communication (a language) that totally depends on our abilities to develop it.

From this point of view Mathematics is an open system that can deeply be changed when its paradigm is changed, and paradigm is not an actual object, but only a theory.

Please read again my answer to Hurkyl:
---------------------------------------------------------------------------
I am not talking about strings or representations.

I am talking about the difference between theory of x and x.

x=x is a tautology, and this point of view does not distinguish between x and model(x).

Mathematics is a theory, therefore any x=model(x).

By this approach the right framework is {model(x)} so no actual x is involved.

When any x is model(x) no x property has an impact on the existence of the framework itself.

for example:

x=model(nothing)

x=model(something)

or if you like:

x=theory(nothing)

x=theory(something)


In both cases x is a theory of x, and we can avoid the paradoxes that caused by x=x point of view.
---------------------------------------------------------------------------

The ZF axiom of the empty set does not see mathematics as theory, but
look at its conclusions as actual results, for example:

If x is not empty then x symbol exists.

If x is empty then x symbol does not exist.

This is a primitive and definitely not an abstract approach.

The same is for the transfinite system, that forcing the theory of infinity to some actual object that its cardinality can be captured by forcing "for all" on the theory(=model) of infinity.

When forcing "for all" on the theory of infinity, you have no theory but actual infinity like "emptiness itself" or "fullness itself".

Both states are beyond any theory, therefore they are the limits of any theory.

Shortly speaking, no information can be exchanged between the actual infinity and the theoretical infinity.

The form of theoretical infinity cannot be but "infinitely many elements".

No infinitely many elements can be an actual infinity.

Your precise Math do not understand this, therefore it is using "for all" on theoretical infinity and the result is like driving by using "full gas in neutral".

 

[matt grime]

Oops, committed some more there haven't you? Like confusing infinite and infinity again.

 

[Organic]

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

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

In both cases standard Math using the word "Quantity".

Emptiness(=no information) or Fullness(=total information)
have no meaning through quantitative point of view.


Also in your subset of Natural numbers that are uninteresting
there is no paradox, because S = {}.


( By the way can you answer to https://www.physicsforums.com/showthread.php?s=&threadid=13461&perpage=12&pagenumber=2
in 3n+1 problem? )

 

[matt grime]

Organic, in simple English, not maths, infinite is an adjective, and infinity is a noun. You mix up the two at will.

As for the paradox, it is only in YOUR opinion that there are no 'uninteresting' numbers. I didn't define what interesting meant so how can you possibly tell me what is or isn't interesting?

 

[matt grime]

As for the request to reply to the other topic - no I don't see the need to. If you'd read the damn link you gave you'd answer your own question.

 

[matt grime]

New technique:

Organic, you use the axiom of infinity induction, what is this? If you are claiming to use the axiom of an infinite set, then explain why, because it just states that an inductive set exists.

Until you define it, or justify it, I will post this question to every reply.

 

[matt grime]

Damn, this is too easy sometimes. But you've either contradicted yourself OR you don't know what a tautology is OR you don't know what = means. You shouldn't have nicked that phrase off me if you didn't know what it meant. You see in the current thread in general math you insist that x=x is only true under certain conditions - either x is actual or theoretical, and not the other. Here you say it is always true. Damn, that was a silly mistake wasn't it?

 

[Hurkyl]

For me Mathematics is only a form of communication (a language) that totally depends on our abilities to develop it.

One would surmise that such a belief would lead to:

(1) a desire to understand what has already been developed.
(2) a desire to learn how to communicate comprehensibly.

Instead, you

(1) make up your own meanings for what has already been developed.
(2) that it is everyone else's problem that they can't comprehend you.

 

[Hurkyl]

You say things like

By not putting any symbol between '{' and '}' we look at emptiness as a constant.

And

I am not talking about strings or representations.

Based on this, I assume you are not referring to ZF at all in this thread. Is this correct?

 

[matt grime]

As often is the case, it is rather unclear what is being talked about, as questions provoke responses that demand more questions.


Here is as synopsis of what we've established.


The axoims of ZF are 'incorrect', in organic's opinion because of his view that the axoim for the empty set contains something unacceptable.

In response one might expect another theory to be put forward. But there isn't one.

The point arises in a semantic, non-mathematical, argument about emptiness, whatever we are supposed to assume that means.

The result is a series of bizarre postings containing little in the way opf recognizable mathematics.

What is clear is that Organic feels the logic of current mathematical thinking is inconsistent. What is also clear is that he doesn't understand much maths, as evinced by his reasoning that 'boolean logic can't cope with infinity', and the following deductions about the requirements of probability, yet he hasn't offered a way to define probability without relying on things he finds inadequate.

So, organic has posted something he states to be a theory for sets, though it seems incomplete. He also doesn't clearly understand what a model is, thinking that the sets we use somehow 'are' ZF, rather than understanding that ZF is a series of rules that our sets obey. There are other set theories out there, each has their own advantages and disadvantages.


For instance, depending on the set theory one uses, one can make different deductions about what the vanishing of ext groups means.

It is quite hard to make sense of it some times, and even harder to make Organic understand what the objections are, especially as I know very little set/model theory.

DOn't know about you, but I feel the goal-posts are constantly shifting in Organic's intents.

 

[Organic]

The logic of actual and potential infinity:

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

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



Emptiness:

E=emptiness

oo...-nor-E-nor-E-nor-E-nor-...oo


Fullness:

F=Fullness

oo...-and-F-and-F-and-F-and-...oo



To use these two logical chains as input, we must break them
but when we break them, we no longer deal with actual infinity
but only with potential infinity of "infinitely many objects".

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

Potential infinity and actual infinity are preventing each other.

Actual infinity cannot be explored by any theoretical method, therefore it is the limit of any theoretical method.

Potential infinity = theory of(actual infinity), no more.

Also fullness and emptiness are preventing each other, but they also define each other as we can see here:

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

By BFC.pdf we can clearly see that Boolean and Fuzzy Logics are private cases of Complementary Logic.

Complementary Logic is based on the symmetry concept and researches its braking levels as natural part of its method.

Therefore numbers are first of all forms of symmetries that are ordered by their internal symmetrical degrees, as we can see hare:

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

and here:

http://www.geocities.com/complementarytheory/P0is1.pdf


Shortly speaking, the Natural numbers by Complementary Logic are Quantum structures, where the standard Natural numbers are only a one private case of some Quantum structure.

 

[Organic]

Hi Hurkyl,

Based on this, I assume you are not referring to ZF at all in this thread. Is this correct?

I take ZF as an example of non fundamental thinking about the set's concept, and I show it by ZF axiom of the empty set (x can be nothing XOR something).

 

[Organic]

Matt,

As for the paradox, it is only in YOUR opinion that there are no 'uninteresting' numbers. I didn't define what interesting meant so how can you possibly tell me what is or isn't interesting?

Yes, you wrote: "The least uninteresting number is intersting"

Therefore S = {}.

 

[matt grime]

'and' and 'nor' are logical operations defined on conditions that are true or false. So E must be a statement that is true of false, it is not a set. I mean the set of natural numbers, in both our worlds, exists, what does it mean for the set of natural numbers to be true (or false) as a statement in logic? I really should have pulled you up about misusing predicates and quantifiers before.


Standard unanswered question: explain what you mean by using the axiom of infinity induction on the list of combinations in new diagonal argument?

 

[matt grime]

Originally posted by Organic
Matt,


Yes, you told: "The least uninteresting number is intersting"

Therefore S = {}.

perhaps i should have more clearly written

the smallest 'uninteresting number' is interesting.

what's your point? How can you conclude that the set is empty? i said the set wasn't empty, i said there were uninteresting numbers.

it's a matter of opinion, that's the problem organic. i mean it might be that just being the smallest uninteresting number isn't interesting, but it might be, who can say what counts??

 

[Organic]

Matt,

Please read this:

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

Take out the relevant part that connected to your question and ask a detailed question about it.

Thank you.

 

[Organic]

Matt,

i said the set wasn't empty

So what.

If the smallest 'uninteresting number' is interesting, then S has no objects in it.

 

[matt grime]

Originally posted by Organic
Matt,

So what.

If the smallest 'uninteresting number' is interesting, then S has no objects in it.

well, done, you're beginning to see the paradox! the set of uninteresting numbers is empty and not empty. do you get it yet?

 

[matt grime]

Originally posted by Organic
Matt,

Please read this:

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

Take out the relevant part that connected to your question and ask a detailed question about it.

Thank you.

Ok, it's still the same question:

how does one get an infinitely long list using 'the axiom of infinity induction'?
that is not a meanignful statement as far as i can tell. this is page 3 paragraph 2. you've never explained this step despite me asking on at least 6 occasions that i can recall off hand.

 

[Organic]

No Matt,

S = {} or S does not exist by your definition.

If it is not empty then by your definition no object is uninteresting.

Therefore S (by your definition) does not exist, because even if one member of it is interesting then we can find the relation of each member in S to this interesting member, so all members are interesting or S has no members.

Shortly speaking no set can include its negation and survive.

 

[Organic]

Do you mean to this part?
--------------------------------------------------------------------------
We can get a combinations list of infinitely many places, by using the ZF Axiom of infinity induction, on the left side of our combinations list, by using the induction on the power_value of each column, for example:

2^0, 2^1, 2^2, 2^3, ...
--------------------------------------------------------------------------

 

[matt grime]

yes i mean that. the axiom of infinity merely tells us that there is something akin to the natural numbers in our model. how does that allow you to produce these strings, and what does it tell you about the strings so produced, and how, for that matter, are you inducting?

 

[Organic]

ZF axiom of infinty simply says: If n exists then n+1 exists.


Do you agree with that?

 

[matt grime]

and what's that got to do with what you're doing?

incidentally, in immediate prelude to that there is a mistake (well, there are many...}

you say the right column is 'based on 2^0' which is questionable, because it is 01010101. check, i think you'll find it's 11001010

 

[Organic]

Matt,

Now I see that you simply don't understand my method.


2^0=1, and 1 tells us after how many times to switch from 0 to 1 or from 1 to 0, therefore the result is 0101010101...

2^1=2, therefore the result is 001100110011...

2^2=4, therefore the result is 0000111100001111...

and so on.

 

[matt grime]

firstly, i think you'll find that the first column in your paper IS NOT 0101010...

now you've explained that, let us examine what this means - given any row it is immediate that after some point, reading right to left, that every entry is zero. This means that the list you construct has only strings that have finitely many non-zero elements in the. Thus it is some sublist of ALL strings. Clearly it is countable.

If you'd explained your method clearly this would be unimportant.

What IS important is you assertion that the list produced contains every element, that is that the diagonal argument produces something that is NOT added to the list because it is already there. Clearly that is not true. Even if only because befoer that you've misused the diagonal argument.

Incidentally, in what way have you USED the axiom of infinity INDUCTION to do anything?

 

[Organic]

Matt.

01010101... 001100110011... 0000111100001111... are columns not rows.

Now I clearly see that you simply don't read what I write in my papers.

 

[matt grime]

look at page 2, the array for n=3 and all the combinations there. you claim that the right hand column is based on 2^1 or something, that is that it ought to be 01010101. it isn't it is 1100101 how can we see that is what you say it is when it clearly isn't?

i know they're columns, where first is the rightmost. look at the rightmost column on page 2 for n=3.

 

[matt grime]

so let us assume that you've picked some ordering of the rows where you've got this pattern - which you CAN do for n=3, as you've got on page 1 and page 3, but not page 2.

So, we can produce a doubly infinite array (to the left and down) where each column is periodic ok, getting there. this is not using the infinity axiom of induction. whatever that might be - you'vr not explained that either, btw, axiom of infinity yes, but not the axiom of infinity induction.

SO? Reading across in each column gives us an element of the combination list, doesn't it? Well, as I stated before, that tells us that each element on the list, that is each row, has only finitely many non-zero entries.

How are you concluding that this list has all the elements you want on it? Cantor's diagonal argument produces something not on that list, it has infinitely many non-zero entries, you claim that this need not be added to the list because it is already on it. No it isn't - this is the basis for you deciding 2^aleph-0=aleph-0

 

[Organic]

Again you show us that you don't fully read what I write, therefore don't understand what you see.

Go back to page 1 read all of it and then read cerfully the first lines in the top of page 2.

 

[matt grime]

Have done, realized what was going on, so see the last post which i reprint here:
so let us assume that you've picked some ordering of the rows where you've got this pattern - which you CAN do for n=3, as you've got on page 1 and page 3, but not page 2.

So, we can produce a doubly infinite array (to the left and down) where each column is periodic ok, getting there. this is not using the infinity axiom of induction. whatever that might be - you'vr not explained that either, btw, axiom of infinity yes, but not the axiom of infinity induction.

SO? Reading across in each column gives us an element of the combination list, doesn't it? Well, as I stated before, that tells us that each element on the list, that is each row, has only finitely many non-zero entries.

How are you concluding that this list has all the elements you want on it? Cantor's diagonal argument produces something not on that list, it has infinitely many non-zero entries, you claim that this need not be added to the list because it is already on it. No it isn't - this is the basis for you deciding 2^aleph-0=aleph-0

 

[Organic]

Be aware that we have an ordered collection of infinitely many 01 sequences by using the ZF axiom of infinity built-in induction on the power level of 2^power_level.

Again ZF axiom of infinity: If n then n+1.

Therefore the power_level = aleph0 and we have an ordered collection of 2^aleph0 elements.

 

[matt grime]

you see, now this is where you go horribly wrong, you pass from a finite cardinal to an infinite one. YOU CANNOT DO THIS! It is not valid.

AND, you've still not said what you mean by

the ZF axiom of infinity in-built induction. What the hell is this?? On the power level 2^powerlevel, what the hell is power level??

And the axiom of infinity does not state if n then n+1

 

[Organic]

Please write the ZF axiom of infinity in English words.

 

[matt grime]

The simple interpretation is that there is an inductive set, that is we can produce a set that behaves like the natural numbers. It is bad english to say that n impies n+1. The maths states that given a set which we can label by n we can form another set which we can label n+1, by taking the union of the set labelled n with the set containing the empty set. It is your english that is most at fault, and virtue of that then your maths. It is the existence of the set indexed by n, with the set containing the empty set that implies there is a set labelled n+1, the labels being the cardinalities.


All you are doing is saying that there infinitely many natural numbers, OK.

let the r'th column of a doubly infinite array be the string of 2^r 0s, then 2^r 1s then 2^r 0s and so on. There is no need to use any axiom of infinity induction (which is...?0 merely that the naturals are infinite! There is no induction! we are not using the s'th level to define the s+1'st level, which is what an induction would be!

 

[matt grime]

Here is a more mathematical definition.


There is a set W that contains the empty set and if any set y is in W then the set containing the union of y and the set containing y is also in . By induction contains every finite integer.


from:

http://www.mtnmath.com/book/node53.html

 

[Organic]

My language is Hebrew.

Now Zf axiom of infinity is:

There is a set Omega that contains the empty set and if any set y is in Omega then the set containing the union of y and the set containing y, is also in Omega.

By induction Omega contains every integer.

So as you see, we are talking about Omega = {1,2,3,...}
and Omega is aleph0.

Therefore our collection is 2^aleph0 collection.

 

[matt grime]

No, aleph-0 is not a set. It is the cardinality of a set. Your omega is the set of natural numbers, its cardinality is Aleph-0.

The last line:

Therefore our collection is 2^aleph0 collection.


Makes little sense to me.

 

[Organic]

http://us.metamath.org/mpegif/aleph0.html

My mistake. Omega = |{1,2,3,...}|


Therefore we have an ordered collection of 2^aleph0 members.

 

[matt grime]

No, you do not have a collection of 2^aleph-0 elements, assuming you mean the set of 'combinations' you produce. I do not know how you are concluding this but it isn't 2^aleph_0. This is the cardinality of the power set of of the natural numbers. The list you produce is not in bijection with any set of cardinality 2^aleph-0, it is in fact obviously of cardinality aleph-0.

 

[Organic]

No Matt,

By using the induction of ZF axiom of infinity on 2^power_value (by mistake i wrote power_level) power_value = aleph0.

 

[matt grime]

edit: word power inserted at crucial point

OK, the cardinality of the power set of n-elements is 2^n, the cardinality of a POWER set of card alpeh-0 is 2^aleph-0. The list you produce has card aleph-0. The only way you could claim it were 2^aleph-0 was if it were the list of all subsets of N, but it isn't it only contains the finite subsets. This is the crux of the issue - you do not produce a 'list' of 2^aleph-0 elements, you prove it is countable yourself.

The only way it you could do otherwise would be to demonstrate the list you produce by some 'induction' that isn't an induction, contains all 'combinations'; it doesn't!

 

[Organic]

Again no Matt,

My collection is all possible 01 combinations of 2^aleph0 ordered elements finite an non-finite.

Please read again page 1 in:
http://www.geocities.com/complementarytheory/NewDiagonalView.pdf

and see for yourself that i am talking about P(aleph0).

 

[matt grime]

But it clearly isn't!

The thing you construct is the doubly infinite array, the first column (right to left) is 0101010...

the second is 0011001100...

the r'th column we now agree is 0000...00111..111000..00... with 2^r 0s and 1s in each segment.

SIMPLE proof this only gives the finite subsets - let x be some combination on the list, say it is the n'th. Since every column after the n'th starts with 2^n zeroes, and 2^n>n for n >1, then clearly every entry after the n'th, reading right to left, is zero, and thus there are only finintely many elements in that subset. THe set was arbitrary.


As you send a string to the binary expansion, to show it's countable, you can only have a finite number of 1s in the damn string!

 

[Organic]

No Matt,

It gives P(aleph0).

Again you don't read what I wrote in page 1 :

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

Your "PROOF" is not about 2^aleph0 members, because n not= aleph0.

 

[matt grime]

But your bizarre bastardization for the finite sets doesn't apply in the infinite list.

In fact as you go top left to bottom right in your diagonal argument, can I ask how you use this in the infinite case - which element in the first string (row) do you use?

I can't say it any more clearly, the list you produce, by your own admission only contains strings with a finite number of non-zero entries on it. This is 'your' proof they are countable (they are trivially countable by construction, but you don't seem to realize this).

By your own admission, the element ..1111 is not on the list, yet it ought to be if the list enumerated the power set. And yet you claim it contains all the combinations... bizarre and self contradictory.

 

[Organic]

Left-right or right-left diagonals holds only for finite P(n).

When I deal with P(aleph0) ordered list then you can see that it is
a right-left diagonal.

It is no problem to say that ...111111 is also in the list but then
we clearly deal with a finite ordered list, which is not our case.

Again, be aware to the fact that we are dealing with an ordered collection.

Also ...111111 is not just a one member but an open interval of aleph0 scales (of 2^aleph0 ordered collection).

 

[matt grime]

Originally posted by Organic
Left-right or right-left diagonals holds only for finite P(n).

When I deal with P(aleph0) ordered list then you can see that it is
a right-left diagonal.

It is no problem to say that ...111111 is also in the list but then
we clearly deal with a finite ordered list, which is not our case.

Again, be aware to the fact that we are dealing with an ordered collection.

Also ...111111 is not just a one member but an open interval of aleph0 scales (of 2^aleph0 ordered collection).

So, we should looka t the diagonal from top right to the 'bottom left'.


the scales thing is not important - ...1111 corresponds to the element in the power set that is the set N.


That doesn't answer anything important anyway.



The thing you construct is a doubly infinite array from right to left and top to bottom, it contains only strings with a finite number of non-zero elements as I've proved independently of you and as you prove yourself by writing an explicit bijection with 2-adic expansions.

Clearly the list is countable (nb, for mathematicians, lists are countable by definition), yet you insist that it contains all combinations, despite proving it doesn't yourself and repeatedly saying the string ..1111 isn't on it! Nor is ..01010101, nor is ..001100110011 etc.

You make two accurate assertions - that there is no bijection between N and P(N) and that the Finite Power set is countable. The problem is you then say they are the same thing! They are not. You prove this yourself.



And I don't understand why you seem think that N is not an element of P(N) {N is the set of natural numbers}, that is the only way I can read your statement about when ...1111 is a combination.

 

[Organic]

1) I proved that |P(N)|>=|N| iff |N| is the cardinal of ALL N members.

2) I clime that there is no such a thing the cardinal of infinitely many elements, because they cannot be completed.

3) In this case all we have is (...111,...000] XOR [...111,...000)

4) There is no shuch a thing [...111,...000]

5) I am going to sleep, so see you and have a good night.