Re: Cantor Diagonal Proof

On 4/14/25 6:43 PM, Lawrence D'Oliveiro wrote:

On Fri, 11 Apr 2025 09:35:15 -0400, Richard Damon wrote:
 

On 4/11/25 3:28 AM, Lawrence D'Oliveiro wrote:

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On Mon, 7 Apr 2025 06:51:02 -0400, Richard Damon wrote:
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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.

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But Cantors arguement wasn't about Computable Numbers ...

 Doesn’t matter. If such a list can be assumed for the purposes of one
proof, it can be assumed for the purposes of another. You can’t argue by
saying it can only be used for purposes that you agree with.

Nope.
Because they are lists of different sorts of numbers.
Cantor begins with the hypothesis of the existance of a complete countable infinite list of the real numbers to prove that such a list does not exist, because we create another number that must be in that original group, but wasn't in the list, so the list must not actually be createable, only hypothesized.
The later proof, starts by proving that there DOES exist a mapping of all the computable numbers to the counting numbers (that might not be computable itself), and then shows that the diagonal must not be computable, as it wasn't on the list of all computable numbers, and thus the diagonal must not be computable, and thus there can't be a master algorithm to compute the initial list.
Thus, the list of computable numbers does exist, but the construction of that list isn't computable, and thus the number on the diagonal must NOT be a computable number.