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From: Richard Heathfield <rjh@cpax.org.uk>
Newsgroups: comp.theory
Subject: Re: Cantor Diagonal Proof
Date: Fri, 4 Apr 2025 20:53:31 +0100
Organization: Fix this later
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On 04/04/2025 20:13, Lawrence D'Oliveiro wrote:
> On Fri, 4 Apr 2025 11:20:28 +0300, Mikko wrote:
> 
>> On 2025-04-04 08:03:44 +0000, Lawrence D'Oliveiro said:
>>
>>> On Fri, 4 Apr 2025 11:00:36 +0300, Mikko wrote:
>>>
<snip>

>>>>
>>>> Can you prove that it computes an incomputable number?
>>>
>>> It’s trying to come up with a number that cannot fit into a set with
>>> cardinality ℵ₀. The cardinality of the computable numbers is the same
>>> as that of the integers, which is ℵ₀.
>>
>> It is also the cardinality of rationals but not of reals. Cantor's proof
>> is that for each list of reals one can construct a real that is not in
>> the list.
> 
> That proof doesn’t quite work, because at any point in the construction,
> the number can be shown to match some later item in the list.

Then show it. I've already explained why you're mistaken, but 
don't let that stop you.

> The mismatch
> with all items in the list is only demonstrated when the proof completes.
> But the proof never completes. Therefore there is no mismatch. QED.

The proof was completed by 1891, when it was published. To claim 
that it "never completes" is like claiming that Achilles never 
catches the tortoise.

-- 
Richard Heathfield
Email: rjh at cpax dot org dot uk
"Usenet is a strange place" - dmr 29 July 1999
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