How Does Complementary Logic Redefine Mathematical Infinity?

[matt grime]

Ok, let us stop using the word complete as it is evidently causing problems,

do you admit to using the infinty axiom of induction on the finite combinations list to construct and infinite one? how do you do this? you do not provide any method for this construction. what do you mean by the axiom of infinity construction? you do use that phrase, as anyone can check:

https://www.physicsforums.com/archive/topic/10675-1.html

you conclude rightly that no (infinitely) long (enumerable) list can contain all the possibly strings of 0s and 1s.

BUT

you then assert that every combination must be on the list (no new combination is produced, in your words). this contradicts what you've just demonstrated. the only way for this to be true is if indeed the list omitted no elements, but you'd have to prove that - you've not done so, cannot do so, and have in fact proved the negation of that statement. why do you insist that you produce no new element, when clearly you do!


the whole argument rests on the assumption that the new element produced by the diagonal argument is on the list, somewhere off the bottom, like the finite cases.

this is wrong, and just you misusing the diagonal argument one of the simplest proofs there is in mathematics. perhaps its simplicity is what people can't cope with.

you map from the list of 01s sends a string to the binary expansion. this can only work for strings with finitely many non-zero elements, so it cannot possibly contain all the strings! you've not got round this problem either.

i'm not sure how i can put it any more plainly than that.

oh, and telling mathematicians cantor is wrong is a good way for them to not listen to anything else you say. even your own favoured Hilbert thought it was right.
hell, it is right.

 

[matt grime]

i'm doing these one point at a time.


if you're going to insist on using probability, then you'd better know that that's an axiomatized system too! in fact for continuous random variables you'll meet countable additivity. bugger, eh?

 

[Organic]

1) aleph0-1 = 2^aleph0 from transfinite point of view.

2) a >= b by Boolean logic cannot be but {}.

3) ...111111111 cannot be in the collection because when it is included in the collection, the collection is complete, and there is no such a thing like: "A complete collection of infinitely many elements".

4) ...111111111 is not just some object but an information structure which exists upon infinitely many scales, and to say that this object is not in the collection, is exactly as if we say that any collection of infinitely many objects has always infinitely many scales to fill.

 

[Organic]

Symmetry and probability have to be used right from the most fundamental level of any mathematical system, and I mean from the level of the set concept, or form the natural numbers.

 

[matt grime]

Originally posted by Organic
1) aleph0-1 = 2^aleph0 from transfinite point of view.

No.

2) a >= b by Boolean logic cannot be but {}.

No. one is a set the other isn't.

3) ...111111111 cannot be in the collection because when it is included in the collection,
the collection is complete, and there is no such a thing like: "A complete collection of infinitely many elements".

? There are other strings with infinitely many ones in them than that.

4) ...111111111 is not just some number but an information structure which exists upon infinitely many scales, and to say that this object is not in the collection, is exactly as if we say that any collection of infinitely many objects has always infinitely many scales to fill.

..1111 is not even a number, never mind 'just a number'

I'd not use 'we' when you mean 'just you'

 

[matt grime]

Originally posted by Organic
Symmetry and probability have to be used right from the most fundamental level of any mathematical system, and I mean from the level of the set concept, or form the natural numbers.

Probabiltity requires the assignment to an event of a number between 0 and 1. It therefore belongs to a 'higher' state of thinking than the naturals because to define it rigorously requires the existence of the naturals, then the rationals, then the reals, as well as a set theory already, as a probability measure is defined on subsets of a proability space. It is not prior to sets as it requires sets for its definition. OF course if you're offering some hand waving 'intuitive' notion of probability then you're on your own again.

 

[Organic]

Matt,

I am talking about a mutation in the natural numbers, that changes the Math paradigm about them.

 

[matt grime]

so you weren't talking abour probability then? were you lying, obfuscating, confused or mistaken? Would it be overly scholastic of me to demand you explain why you've chosen the word mutation, adding yet another ill-defined term to the mix? Apart from your evident desire to confuse the issue so that you never explain anything that's asked of you.

 

[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.

 

[matt grime]

Almost never do the words you use have a well-defined mathematical meaning, and when they do it is never in the sense that you are using them.

So, I know what mutation means, now you need to explain what mutation you are talking about. Try and be self-consistent: thus far that has eluded you becaue you keep introducing more concepts that you can't explain before sorting out the previous problems.

Before stating that current maths is wrong, perhaps you should understand what it says? I mean you've demonstrated in the last few posts that you don't know what 'equals' means, nor what probability is, that you don't understand the definition of aleph-0, that you don't know what the valid operations on cardinals are, that you believe there is something called the axiom of infinity induction that we all know and love, and that you can't understand the idea of the proof of cantor's diagonal argument. if you can'it understand these, and because of this lack of understanding decide to write a new theory without defintions, then what are we supposed to do?

 

[Organic]

No Matt,

I started to write my ideas after I understood that pure mathematics have no ability to deal with complex systems.

And the reason is very simple, it does not distinguish between x-model and x.

I say: x=model(x)

And by this approach my theory do not run after its own tail, creating virtual paradoxes like Russell's paradox, forcing words like "complete" and "all" on infinitely many elements and by this, creating and using almost all its energy to explore the results of this forcing, which is the transfinite words.

When you don't understand my results when using the built-in induction of the axiom of infinity on a sequence of 01 notations, you clearly show that you don't have the most important things that a good mathematician needs, which are:
sense of symmetry and sense of simplicity that implies sense of beuty.

Technical abilities are only tools but because a lot of mathematicians like you do not understand the difference between x-model and x, they become closed systems under their own technical conventions and cannot see anything beyond it.

You are a good example of such a mathematician.

Term: ( http://mathworld.wolfram.com/Term.html )
In logic, a term is a variable, constant, or the result of acting on variables and constants by function symbols.

It is nice isn't it?

But standard math does not use the difference between variable and constant when it define the set concept, and the result is:

By the axiom of the empty set we clearly see that x cannot be but a non-empty object(=something), and only then we can define {}.

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

If x is not empty then x symbol exists.

If x is empty then x symbol does not exist.

And you call this approach an abstract Math?

 

[matt grime]

I notice that you do not answer the charge that you misuse notations.

First, the whole point of ZF is that it avoids Russell's paradox.

Simplicity and symmetry have nothing to do with you misusing the axiom of infinty (which merely states that an infinte, indcutive set exists, namely N).

The rest of your post is rambling and incoherent, in particular the assertion that the emtpy set is what you get when you put nothing in between some parentheses.

 

[matt grime]

OH, and given that you apparently cannot even understand the simplest statements about mathematics (Collatz 3n+1 thread), I find you casting aspersions on my abilities just a little offensive. So, explain the difference between the empty set and the model of the empty set.

And, btw, I'd be the first to say I'm not a good mathematician.

 

[Hurkyl]

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

Again, it sounds like you're talking about lexical analysis, not set theory. The null string is indeed a constant in that context. And, of course, if ξ is a variable representing a string, then ξ may certainly be the null string, and if we have a proposition that looks like $\forall \xi: P(\xi)$ (where ξ ranges over strings), then $P(\epsilon)$ must be true (where I've chosen to use the symbol ε to denote the null string).

But do beware; when ξ is a variable representing a string, the string '{ξ}' does not denote the set {ξ}. (The former is the string denoted by ξ surrounded by braces, and the latter is the set containing the string ξ)

For a concrete example, the string

'{a, b}' is not the set {'a, b'}. (where a and b are not string variables) The former is a string that denotes the two element set {a, b}, while the latter is a set containing the string 'a, b'.

 

[Organic]

Hi 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 distingiush 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.

 

[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.