Is the proof of the following theorem correct?

[Mario Carneiro]

Without commenting on the proof, there is an odd consequence of the
statement: Condition 3 says that A(k,s) halts, so condition 1 says that
C_k(s) diverges, so the second part of condition 2 asserts that C_s
diverges everywhere. That's not an error as far as I can tell, but it does
suggest that some part of the proof might be unexpectedly trivialized.

Mario Carneiro

On Tue, Jul 20, 2021 at 8:33 PM X.Y. Newberry <newberryxy at gmail.com> wrote:

> For FOM:
>
> Is the proof of the following theorem correct?
>
> THEOREM 1: There are numbers k and s and a program A(n,m) satisfying the
> following conditions.
>
> 1. If A(n,m) halts, then C_n(m) diverges.
> 2. For all n, C_k(n) = A(n,n) and C_s(n) = Ck(s).
> 3. A(k,s) halts and for all n, A(s,n) diverges.
>
> Here C_n(*) is a program with index n in some exhaustive enumeration of
> all possible programs.
>
> The proof is here.
> https://xnewberry.tripod.com/Theorem.pdf
>
> --
> X.Y. Newberry
>
> *There are two ways to be fooled. One is to believe what isn't true; the
> other is to refuse to believe what is true.*
> ― Søren Kierkegaard
> <https://www.goodreads.com/author/show/6172.S_ren_Kierkegaard>
>
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