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From: Richard Damon <richard@damon-family.org>
Newsgroups: comp.theory
Subject: Re: Cantor Diagonal Proof
Date: Fri, 11 Apr 2025 09:35:15 -0400
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On 4/11/25 3:28 AM, Lawrence D'Oliveiro wrote:
> On Mon, 7 Apr 2025 06:51:02 -0400, Richard Damon wrote:
> 
>> Your problem is you assume you can compute the nth value from the value
>> of n, but that requires you master algorithm include an infinite number
>> of algorithms in itself to choose from to build that number.
> 
> But the Cantor construction assumes you can construct that list. So if you
> object to the assumption of the existence of such a list, then you knock
> down Cantor’s proof as well.

But Cantors arguement wasn't about Computable Numbers, so the method of 
construction doesn't need to be a computation.

We CAN "Construct" the list by an algorithm that can handle the 
non-finite, which means it isn't a computation.

This seems to be a root of your problem, that you are confusing the 
later adaptation of the logic of Cantor's proof to show that the 
Cantor's argument does not apply to Countable Numbers.

So, in one sense you are right to think that it can't be showing that 
there is a way to compute a number on the diagonal showing that the 
computable numbers are not countable, because that *IS* the result of 
that proof, that we can't use Cantor's Diagonal to show that the 
Computable Numbers are not Countable, since we can prove otherwise that 
they are.

We CAN built an ordered list of all computable numbers (but not compute 
that list), but since we can't compute the list, we can't compute the 
diagonal. (We can't compute the list as that would require solving the 
halting problem)

The problem is this doesn't apply to the Cantor's actual proof which 
wasn't at all about computability, so the "construction" of the diagonal 
doesn't need to be a computation.