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From: Richard Damon <richard@damon-family.org>
Newsgroups: comp.theory
Subject: Re: Cantor Diagonal Proof
Date: Fri, 11 Apr 2025 09:24:57 -0400
Organization: i2pn2 (i2pn.org)
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On 4/11/25 3:26 AM, Lawrence D'Oliveiro wrote:
> On Thu, 10 Apr 2025 21:21:18 -0400, Richard Damon wrote:
> 
>> On 4/10/25 8:28 PM, Lawrence D'Oliveiro wrote:
>>>
>>> On Wed, 9 Apr 2025 22:00:11 -0400, Richard Damon wrote:
>>>
>>>> But a finite list can't get you to the needed arbitrary precision
>>>> needed.
>>>
>>> I was going to say, sure it can, because the size of the list is a
>>> function of the precision you ask for.
>>
>> But the function needs to be prepared to handle ANY precision, and thus
>> needs to be infinite.
> 
> That’s true of computable numbers in general, so unless you’re objecting
> to the very existence of the concept, it’s still irrelevant.
> 
>> But you need to remember that he wasn't "constructing" it in the manner
>> you are assuming, it isn't being constructed by a finite function, as
>> that wasn't the domain he was talking about.
> 
> Given the example list I gave elsewhere, there is a fundamental conflict
> between a proof by induction (a well-established technique) and a proof by
> his construction (which has to be seen as something novel). I would say
> that points to a logical weakness in his construction.

Remember, Cantor wasn't talking about "Computable" numbers, his list was 
a proported list of all reals mapped one to one with the Natural 
Numbers, something doable if they were countable.

The problem with your "induction" is you assumed the existance of a 
computation that doesn't exist, a computation that given the nth digit 
of the nth computable number for any value of n.

Such a construction, by necessity to work for ALL n, must have an 
infinite algorithm, and thus can't be just assumed to exist.