My Math Forum > Math I don't see the significance of Cantor's diagonal argument

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January 11th, 2017, 04:29 PM   #21
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Quote:
 Originally Posted by Maschke Didn't v8archie explain that all your decimals have only finitely many nonzero digits, hence are rational? Where is .101010101010101010101... on your list?
But you said that everything exists at once. An infinite number of natural elements would have constructed 0.101010... somewhere, no? If we had infinite time and energy, I could physically show you where 0.101010... is.

Or is "going to infinity" and "exhausting infinity" different things? If so, then that would make sense to me about why my method wouldn't work.

January 11th, 2017, 04:34 PM   #22
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Quote:
 Originally Posted by Mathbound But you said that everything exists at once. An infinite number of natural elements would have constructed 0.101010... somewhere, no? If we had infinite time and energy, I could physically show you where 0.101010... is.
Sure, all the real numbers exist. But to show that they're countable, you have to exhibit a bijection between them and the natural numbers. You haven't done that.

I apologize if my explanation added confusion.

January 11th, 2017, 04:38 PM   #23
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Quote:
 Originally Posted by Mathbound Because we can exhaust all decimal places for a number like pi - 3 using natural numbers (or aleph 0 elements), I don't see how my list a few posts above this post could leave out a real number. Like, I do and I don't see. On one hand, the rule forces us to leave out a number. But on the other hand, my list gives every possible digit in every possible decimal place. It will eventually give all possible combinations of digits 0 to 9 in all decimal positions for real numbers between 0 and 1.
Your list starts with decimals of one digit before infinite zeros, then two before infinite zeros, then three, four, five,...

Every number on the list has a finite (natural) number of digits followed by infinite zeros. Specifically, it contains no decimals that do not end in infinite zeros such as $(\pi-3)$, $(\sqrt2 -1)$ and $\frac13$.

The only way your list could move onto numbers that do not end in infinite zeros would be for the process of listing all decimals with $N$ digits before the infinite zeros to halt (where $N$ is some natural number). This means that no number having $(N+1)$ digits followed by infinite zeros would appear on the list. So it's still not a complete list.

January 11th, 2017, 04:44 PM   #24
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Quote:
 Originally Posted by Mathbound Or is "going to infinity" and "exhausting infinity" different things?
"Going to infinity" has no meaning in this context. The number of digits before the infinite zeros follows the sequence 1, 2, 3,... This sequence never reaches "infinity" because "infinity" is not a number. So each number on your list has a finite number of digits before the infinite (that is "never ending") sequence of zeroes.

January 11th, 2017, 04:54 PM   #25
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Quote:
 Originally Posted by Maschke Sure, all the real numbers exist.
This is a debatable assertion. Real numbers "exist" as a mathematical construct. They cannot be demonstrated to "exist" in the physico-temporal world. Nor can any transfinite number be shown to have a physico-temporal existence.

It would, I think. be preferable to say something like "if you accept that real numbers and at least one transfinite numbers 'exist' as mathematical constructs, then it follows that the (transfinite) number of the real numbers exceeds the (transfinite) number of the natural numbers."

I suspect that the psychological barriers to grasping Cantor include the refusal to entertain seriously the "existence" of transfinite numbers and the failure to realize that the "existence" of transcendental numbers entails the "existence" of transfinite numbers. Cantor's argument is elegant and relatively simple once you accept its ontological premises. But those premises can never be "proved" physically.

January 11th, 2017, 05:02 PM   #26
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Quote:
 Originally Posted by JeffM1 This is a debatable assertion. Real numbers "exist" as a mathematical construct. They cannot be demonstrated to "exist" in the physico-temporal world. Nor can any transfinite number be shown to have a physico-temporal existence. It would, I think. be preferable to say something like "if you accept that real numbers and at least one transfinite numbers 'exist' as mathematical constructs, then it follows that the (transfinite) number of the real numbers exceeds the (transfinite) number of the natural numbers." I suspect that the psychological barriers to grasping Cantor include the refusal to entertain seriously the "existence" of transfinite numbers and the failure to realize that the "existence" of transcendental numbers entails the "existence" of transfinite numbers. Cantor's argument is elegant and relatively simple once you accept its ontological premises. But those premises can never be "proved" physically.
Of course. The remark that you quoted was based on a comment I made in #11:

Conceptually it's like a lookup table or array in a computer program; except that the array has infinitely many cells, one for each natural number; and we can look up any cell of the array immediately. That's the conceptual model of standard math. Everything exists at once and is always available for use. That's one of the basic rules of the game. As you can see this is a highly abstract conceptual world we live in when we do math.

That para was a replacement for an extended discussion of the role of the Axiom of Infinity, which I wrote in an earlier draft of #11 and then deleted as being inappropriate to the OP's question.

But for the record, Cantor's great breakthrough was to go past Aristotle and posit the existence of a completed infinity of natural numbers. Without that, there's no transfinite theory at all.

Last edited by Maschke; January 11th, 2017 at 05:09 PM.

 January 11th, 2017, 06:30 PM #27 Member   Joined: Jun 2014 From: Alberta Posts: 55 Thanks: 2 Thanks everyone, I think I get it now.

 Tags argument, cantor, diagonal, significance

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