# A new point of view on Cantor's diagonalization arguments

by Organic
Tags: arguments, cantor, diagonalization, point, view
 Sci Advisor HW Helper P: 9,398 Organic would need to demonstrate the the natural numbers are not sufficient for the job they are used for for that parallel to hold here. The greeks thought that all numbers on their number line were rational originally. This is not true, therefore their assumption is wrong. Where do we go wrong with the definition that the natural numbers are the basic counting blocks of finite sets? What has gone wrong? Sure we can add all the extra structure to it you want, but it doesn't alter the basic facts. You could argue the 'inventing' the negative integers, then the rationals, then the reals and then the complexes are all paradigm shifts, but they don't mean the naturals are incomplete within themselves. We dont' assume those are all the arithmetical objects that exist. As I keep telling organic he's just playing around with basic operads and monads, if he wants to add extra structure to them that's fine, but he's already using the naturals inside his definition to define his "new naturals" there is nothing there to suggest these new objects should replace the natural numbers. That is not the objection to what he's doing. The objection is that he's making wild unsubstantiated and plainly false claims about things he doesn't understand.
 P: 262 Matt: You Ignore by your question to Organic all my background that i share with you here as i promise you few hour ago! Moshek I will let Organic to answer your question about his numbers. [:))]
P: 1,210
Matt,

First, you have to show me that you understand what I am doing.

Each one of these structural-quantitative products is unique, therefore can be used as a building-block for much more interesting and richer information form, then your “quantitative-only" unique [n] result, which is nothing but a private-case of no-redundancy-no-uncertainty structural-quantitative product of my number system.

We can clearly see this here:

http://www.geocities.com/complementa...ry/ETtable.pdf

Matt,

Natural number system of Standard Math is not simple but trivial, and my attitude is to give it the power of simplicity instead of the weakness of triviality.

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http://plato.stanford.edu/entries/qm-copenhagen/#4

This paradigm's shift, does not exist in the basis of Standard Math language, because Boolean Logic or Fuzzy Logic are private cases of what I call Complementary Logic, that an overview of it can be found here: http://www.geocities.com/complementarytheory/BFC.pdf

Through my point of view Natural numbers are complementary elements, based on discreteness(particle-like)-continuum(wave-like) associations.

The information structure of the standard Natural numbers, is only a private case of these associations, for example:

http://www.geocities.com/complementa...ry/ETtable.pdf

More details can be found here:

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

Man is no longer an observer but a participator, which its influence must be included in any explored system.

It mean that we cannot ignore our cognition's abilities to create Math language anymore, as I clearly show here:

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

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 The quantum shift is mathematical: quantum groups, quantum cohomology, quantized universal enveloping algebras, quantum chaos, even quantum mechanics (I learned that in a maths degree...), I've even been learning about quantum linear algebra. And then there's quantum computing and quantum information theory, both well established and mathematical. And did I mention quantum Kac-Moody Lie algebras? Deformation Theory? q-schur algebras, quantum binomial coefficients?
Please show me some influence on basic thing like the natural number, by one of these theories.
P: 1,210
 but he's already using the naturals inside his definition to define his "new naturals" there is nothing there to suggest these new objects should replace the natural numbers.
So, you dont understand that the stantard natural number is a trivial private case of infinitely many structural/quantitative information's forms that ignored by Standard Math paradigm.

Do you get it?

A NUMBER is first of all an information's form, and to understand this we MUST explore our cognition's abilities to define this information's form, as I do here:

http://www.geocities.com/complementarytheory/count.pdf
HW Helper
P: 9,398
 Originally posted by Organic Each one of these structural-quantitative products is unique, therefore can be used as a building-block for much more interesting and richer information form, then your “quantitative-only" unique [n] result, which is nothing but a private-case of no-redundancy-no-uncertainty structural-quantitative product of my number system.
This would only hold true if I had ever said that the Natural Numbers are the only thing in mathematics. We have N, simply, N, the set of natural numbers. Then there is (N,+) which is a semi-ring. Then the is (N,<) the ordinals. And who can forget (N,+,<) the ordered semi-ring. On top of that one might do (N[-1],+), ie Z under adding formal inverses. Not forgettiing the ordering too. What about localizing and getting a field, Q, and completing to get R, and closing to get C? Then there's the division rings and Wedderburns' structure theorem over that.

Of course there is the general theory of semi-rings etc that 'has N as a private case of a larger language', in you words.

There is a whol rich tapestry of objects that have N, Z, Q or R as the simplest version. This apparent paradigm shift is already there.

If you incidentally knew about the quantized structures above and the rigorous framework behind quantum mechanics, then perhaps you'd understand the analogies.
 P: 1,210 Matt, Two critical things you ignore when you define N members: 1) A research of your ability to count: http://www.geocities.com/complementarytheory/count.pdf 2) that natural numbers are first of all information forms, therefore Their minimal existence must start form here: (1*4) ={1,1,1,1} <------------- Maximum symmetry-degree, ((1*2)+1*2) ={{1,1},1,1} Minimum information's (((+1)+1)+1*2) ={{{1},1},1,1} clarity-degree ((1*2)+(1*2)) ={{1,1},{1,1}} (no uniqueness) (((+1)+1)+(1*2)) ={{{1},1},{1,1}} (((+1)+1)+((+1)+1))={{{1},1},{{1},1}} ((1*3)+1) ={{1,1,1},1} (((1*2)+1)+1) ={{{1,1},1},1} ((((+1)+1)+1)+1) ={{{{1},1},1},1} <------ Minimum symmetry-degree, Maximum information's clarity-degree (uniqueness) After these two "MUST HAVE" steps, we can continue to develop the next numbers in the number system.
 Sci Advisor HW Helper P: 9,398 Why are these must have steps? We can define the natural numbers without them, so they obviously aren't must have are they? Or are you forgetting the DEFINITION of the natural numbers (as a set, just a set, with no other structure)?
P: 1,210
Matt,

Any number is first of all an information form, therefore any aspect of information form MUST be researched by us, where our cognition’s abilities to research information MUST be included too.

Form this point of view, redundancy AND uncertainty cannot be ignored, and through this approach(which is not an extra approach but the MINIMAL approach to understand the natural number concept) we can clearly show that the standard natural numbers are only a one and only one private case of verity of information forms, which are ordered by their vagueness degrees from maximum vagueness to minimum vagueness when a given quantity remains unchanged.

Man is no longer an observer but a participator, which its influence must be included in any explored system.

The above is the QM paradigm shift that is not understood yet by the current community of pure mathematicians.

For example: Be aware that what you call a function is first of all a reflection of your memory.

 Or are you forgetting the DEFINITION of the natural numbers (as a set, just a set, with no other structure)?
A set is only a framework that helps us to explore our ideas, no less no more.

When there is a paradigm shift this framework is chaneged too.
 Sci Advisor HW Helper P: 9,398 It would be nice if you told us what the uncertainty and redundancy of a number is. So that we know what we MUST be aware of.
 P: 1,210 The answer is here: http://www.geocities.com/complementarytheory/count.pdf
HW Helper
P: 9,398
 Originally posted by Organic The answer is here: http://www.geocities.com/complementarytheory/count.pdf
Ah, of course! The answer to my question is in another pdf! This one doesn't even mention the words uncertainty and redundancy. In what way is that an answer?

Do you mean to imply that numbers are uncertain because if we have set of identical objects we cannot distinguish between them? I don't see why that makes numbers uncertain. I see why it makes identifying identical objects impossible, but that has nothing to do with quantity. Suppose I just had one bead. I blink, I see an identical bead in the same place. is it the same bead? I do that for more than 1 bead.. what has quantity got to do with it?
 P: 1,210 From your response it looks that you did not read the pdf file. Please reply if you have some technical problems to open the pdf. Thank you, Organic
 Sci Advisor HW Helper P: 9,398 I opened the pdf link you gave and didn't see the word uncertainty mentioned once, please tell me which line it is on. Or, as the pdf is only a page long, try stating in plain simple English what you mean the the uncertainty of a number, or its redundancy. It's another simple request. Actually it's the same simple request isn't it? A simple paragraph starting: The redundancy of a number is... or perhaps it ought to start A number is redundant if... or even, given a number n, and some object related to n, then the object is redundant if...
 P: 1,210 Dear Matt, You are going too far. This pdf is not a technical paper, but a simple test that answer to the question: what are the minimal conditions that give us the ability to count? Please read it again from this point of view, and don't search for any definitions there, just try to understand this simple test. Thank you. Organic
 P: 5 Cantor's Diagonalization Argument Theorem-The interval [0,1] is not countably infinite. Proof:-The proof by contradiction proceeds as follows: Assume (for the sake of argument) that the interval [0,1] is countably infinite. We may then enumerate all numbers in this interval as a sequence, ( r1, r2, r3, ... ) We already know that each of these numbers may be represented as a decimal expansion. We arrange the numbers in a list (they do not need to be in order). In the case of numbers with two decimal expansions, like 0.499 ... = 0.500 ..., we chose the one ending in nines. Assume, for example, that the decimal expansions of the beginning of the sequence are as follows: r1 = 0 . 5 1 0 5 1 1 0 ... r2 = 0 . 4 1 3 2 0 4 3 ... r3 = 0 . 8 2 4 5 0 2 6 ... r4 = 0 . 2 3 3 0 1 2 6 ... r5 = 0 . 4 1 0 7 2 4 6 ... r6 = 0 . 9 9 3 7 8 3 8 ... r7 = 0 . 0 1 0 5 1 3 5 ... ... We shall now construct a real number x in [0,1] by considering the kth digit after the decimal point of the decimal expansion of rk. r1 = 0 . 5 1 0 5 1 1 0 ... r2 = 0 . 4 1 3 2 0 4 3 ... r3 = 0 . 8 2 4 5 0 2 6 ... r4 = 0 . 2 3 3 0 1 2 6 ... r5 = 0 . 4 1 0 7 2 4 6 ... r6 = 0 . 9 9 3 7 8 3 8 ... r7 = 0 . 0 1 0 5 1 3 5 ... ... The digits we will consider are indicated in bold and illustrate why this is called the diagonal proof. From these digits we define the digits of x as follows. if the kth digit of rk is 5 then the kth digit of x is 4 if the kth digit of rk is not 5 then the kth digit of x is 5 For the example above this will result in the following decimal expansion. x = 0 . 4 5 5 5 5 5 4 ... The number x is a real number (we know that all decimal expansions represent real numbers) in [0,1] (clearly). Hence we must have rn = x for some n, since we have assumped that ( r1, r2, r3, ... ) enumerates all real numbers in [0, 1]. However, because of the way we have chosen 4's and 5's as digits in step (6), x differs in the nth decimal place from rn, so x is not in the sequence ( r1, r2, r3, ... ). This sequence is therefore not an enumeration of the set of all reals in the interval [0,1]. This is a contradiction. Hence the assumption that the interval [0,1] is countably infinite must be false. Q.E.D. It is a direct corollary of this result that the set R of all real numbers is uncountable. If R were countable, we could enumerate all of the real numbers in a sequence, and then get a sequence enumerating [0, 1] by removing all of the real numbers outside this interval. But we have just shown that this latter list cannot exist. Alternatively, we could show that [0, 1] and R are the same size by constructing a bijection between them.
 Emeritus Sci Advisor PF Gold P: 5,540 Cantormath, Welcome to Physics Forums! And you're right, there is no argument. What you've stumbled upon here is a piece of crackpottery from the old days when we adopted an "anything goes" attitude in the Theory Development Forum. We've since tightened things up so that we only allow things that make sense. So feel free to stick around and enjoy the Forums. Don't worry about this thread, because the orginal poster isn't even here anymore.

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