Re: Cantor Diagonal Proof

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.