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From: Richard Heathfield <rjh@cpax.org.uk>
Newsgroups: comp.theory
Subject: Re: Cantor Diagonal Proof
Date: Thu, 10 Apr 2025 01:51:07 +0100
Organization: Fix this later
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On 10/04/2025 01:41, Lawrence D'Oliveiro wrote:
> On Tue, 8 Apr 2025 18:57:17 -0400, Richard Damon wrote:
> 
>> On 4/8/25 5:02 PM, Lawrence D'Oliveiro wrote:
>>>
>>> On Mon, 7 Apr 2025 18:24:36 -0400, Richard Damon wrote:
>>>
>>>> And that is because while every element on the list has an algorithm
>>>> to construct it, that list is infinite, so you can't just put them
>>>> *ALL* in to one finite algorithm to compute any one you need at the
>>>> moment.
>>>
>>> But that’s exactly how computable numbers work.
>>
>> No, *A* computable number has a finite algorithm that computes it.
>> Finite in having finite instructions in its algorithm and finite states
>> to process.
>>
>> The problem is your "master" algorithm need the algorithms of *ALL* the
>> computable numbers within it, which is an infinite number of algorithms,
>> and thus isn't itself a finite algorithm.
> 
> That is the problem with the Cantor construction, not with my disproof of
> it. My disproof of it only needs a finite number of list elements at any
> point in the proof. The Cantor construction needs them all.

The Cantor argument doesn't need /any/ of them.

Richard Damon rightly argues that you can't construct a complete 
list of computable numbers. Cantor argues that even if you think 
you've succeeded, the Cantor diagonal shows you one you missed, 
and it doesn't need the list to be complete just to demonstrate 
that it isn't.


-- 
Richard Heathfield
Email: rjh at cpax dot org dot uk
"Usenet is a strange place" - dmr 29 July 1999
Sig line 4 vacant - apply within