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Re: Cantor Diagonal Proof

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The message body is Copyright (C) 2025 Tristan Wibberley except
citations and quotations noted. All Rights Reserved except as noted in
the sig.

On 22/10/2025 14:29, Richard Heathfield wrote:
> On 22/10/2025 13:40, Tristan Wibberley wrote:

> Cantor's diagonal argument proves that uncountable sets exist. At no
> point did he claim that the number the argument computes is incomputable.
> 
>> I'm still unconvinced that he successfully [even defined the crucial number].
> 
> He didn't try. An existence proof doesn't have to define a number.

True. I was being loose. It would have to construct every list itself.

I suppose I should have said "Doesn't Cantor's argument merely show that
the number is definable contingent on that the function from a list to
the crucial number, if I may call it that, is defined for every list?"
Can you advise some succinct alternative using contemporary terms-of-art
instead of that mouthful?

If there is /any/ list from which the function that defines the crucial
number is not defined then there is a list for which one hasn't shown,
absent further argumentation, that a number is missing from the list.


>> take these descriptions:
>>
>>    f_0 is the defining sequence of numbers
>>
>>    g is the sequence of generated "new" numbers, generated as they
>>    say cantor described (I haven't read his work directly)
>>
>>    f_1 is some choice of numbers
>>
>>    even(n) and odd(n) are as you'd normally expect
>>
>>
>> then I can provide an f_0:
>>    where f_0(n) = case n of
>>              even(n) -> f_1(n/2)
>>              odd(n) -> g((n-1)/2)
>>
>> here, every generated "new" number is eventually included in the list.
> 
> And yet your list remains incomplete.

It remains partially specified. I've defined a class of lists for each
of which Cantor's crucial missing number is not shown to exist by his
method.

If it is to be proved incomplete, it will be via a different argument
than what I have been, to date, told is Cantor's.


>> To show that the set of reals is larger than the set of naturals one
>> would have to show that there's no definition of f_1 such that it
>> contains all the reals!
> 
> No, you'd only have to show that you can't place the reals in one-to-one
> correspondence with the integers.

Then there would be no definition of /f_1/ such that it contains all the
reals and we have begged the question, demonstrating that Cantor's
argument (what I understand it to be from several educators and
wikipedia) is insufficient.

Since I've partially fulfilled Cantor's argument structure sufficiently
to preclude the existence of lim{n->inf} of g(n). His argument relies on
that, doesn't it, even if he didn't use that notation or its equivalent
prose. His argument can't show that the extra number exists for /all/
lists because it doesn't exist however you may suppose f_1 to be
defined. At best his argument can show that an extra number is not in
the list for /some/ lists... those lists that do not include the numbers
of their respective g.

Since I can do that for f_1, and then every f_k, I need a /different/
argument strategy to show that there's no such one-to-one correspondence.

I have begged the question within Cantor's argument strategy, haven't I,
by finding a class of lists for which there is no number that Cantor's
method shows to be missing from any of its members. There is a
/different/ argument that /is/ a proof if the conclusion holds.


An assertion of an axiom that there's no one-to-one correspondence (as
you provided) would be an example of a different argument and I thank
you for the succinct statement of such an axiom that you provided. If it
is an axiom, though, I don't think Cantor's argument for it is of any
interest and if he relies on it as an alternative to taking a limit then
I don't think his doctrine is even an argument.

--
Tristan Wibberley

The message body is Copyright (C) 2025 Tristan Wibberley except
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Thread

Re: Cantor Diagonal Proof Tristan Wibberley <tristan.wibberley+netnews2@alumni.manchester.ac.uk> - 2025-10-22 13:40 +0100
  Re: Cantor Diagonal Proof Richard Heathfield <rjh@cpax.org.uk> - 2025-10-22 14:29 +0100
    Re: Cantor Diagonal Proof Tristan Wibberley <tristan.wibberley+netnews2@alumni.manchester.ac.uk> - 2025-10-22 16:02 +0100
      Re: Cantor Diagonal Proof Julio Di Egidio <julio@diegidio.name> - 2025-10-22 17:17 +0200
      Re: Cantor Diagonal Proof Tristan Wibberley <tristan.wibberley+netnews2@alumni.manchester.ac.uk> - 2025-10-22 17:26 +0100
    Re: Cantor Diagonal Proof WM <wolfgang.mueckenheim@tha.de> - 2025-10-23 16:51 +0200
  Re: Cantor Diagonal Proof Julio Di Egidio <julio@diegidio.name> - 2025-10-22 16:19 +0200
    Re: Cantor Diagonal Proof Tristan Wibberley <tristan.wibberley+netnews2@alumni.manchester.ac.uk> - 2025-10-22 17:32 +0100
    Re: Cantor Diagonal Proof WM <wolfgang.mueckenheim@tha.de> - 2025-10-23 16:56 +0200
      Re: Cantor Diagonal Proof: un-_Countable_ or un-_Cartesian_? Ross Finlayson <ross.a.finlayson@gmail.com> - 2025-10-23 12:27 -0700
  Re: Cantor Diagonal Proof "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2025-10-22 12:15 -0700
  Re: Cantor Diagonal Proof wm <wolfgang.mueckenheim@tha.de> - 2025-10-23 16:34 +0200
    Re: Cantor Diagonal Proof Tristan Wibberley <tristan.wibberley+netnews2@alumni.manchester.ac.uk> - 2025-10-23 20:43 +0100
      Re: Cantor Diagonal Proof WM <wolfgang.mueckenheim@tha.de> - 2025-10-24 22:35 +0200

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