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A new point of view on Cantor's diagonalization arguments 
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#163
Mar2304, 10:33 AM

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. 


#164
Mar2304, 10:37 AM

P: 1,210

Matt,
First, you have to show me that you understand what I am doing. For example, it is the second time that you write general things about my work instead of reading and and reply detailed comments on this: Each one of these structuralquantitative products is unique, therefore can be used as a buildingblock for much more interesting and richer information form, then your “quantitativeonly" unique [n] result, which is nothing but a privatecase of noredundancynouncertainty structuralquantitative 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.  The big paradigm's shift is QM and not SR, please read this: http://plato.stanford.edu/entries/qmcopenhagen/#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(particlelike)continuum(wavelike) 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  


#165
Mar2304, 10:47 AM

P: 1,210

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 


#166
Mar2304, 03:17 PM

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Of course there is the general theory of semirings 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. 


#167
Mar2304, 03:34 PM

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:



#168
Mar2304, 05:55 PM

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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)?



#169
Mar2404, 01:59 AM

P: 1,210

Matt,
The answer is very simple. 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. When there is a paradigm shift this framework is chaneged too. 


#170
Mar2404, 04:01 AM

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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.



#171
Mar2404, 04:24 AM

P: 1,210



#172
Mar2404, 05:11 AM

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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? 


#173
Mar2404, 06:07 AM

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 


#174
Mar2404, 09:39 AM

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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... 


#175
Mar2404, 09:57 AM

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 


#176
Oct1104, 03:49 AM

P: 5

Cantor's Diagonalization Argument
TheoremThe 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. 


#177
Oct1104, 03:07 PM

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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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