Re: Cantor Diagonal Proof

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.