Re: Cantor Diagonal Proof

On 4/10/25 8:28 PM, Lawrence D'Oliveiro wrote:

On Wed, 9 Apr 2025 22:00:11 -0400, Richard Damon wrote:
 

On 4/9/25 8:41 PM, Lawrence D'Oliveiro wrote:

>
On Tue, 8 Apr 2025 18:57:17 -0400, Richard Damon wrote:
>

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.

>
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.

 But that’s irrelevant, because if you’re saying an infinite list cannot be
constructed, then that blows the Cantor construction out of the water,
since an infinite list is precisely what it assumes.

But Cantor wasn't talking about "Construction" in that manner, as he wasn't talking about computability.
He was describing a method for a ideal infinite logic to create the value.

 

Not also, Cantor only claimed that the numbers weren't countable, he
didn't talk about computable. That is a later work looking at the proof.

 True. But interesting things happen when you try to apply his construction
to other kinds of number lists.

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.