Let $C$ be a conjecture about natural numbers. Let $$S = \{n\in N: n > m \text{ where $m$ is the first number found for which $C$ is false} \} $$
Is $S$ decidable?
If $C$ is true for all natural numbers then S is an empty language which is decidable.
If $C$ is false, then m exists and $S=\{m+1, m+2, ...\}$
Let $M_k$ be a machine that accept numbers $> k$ and rejects numbers $\le k$.
I don't know which $M_k$ decides $S$, but I know one of them does. Is this sufficient to prove that $S$ is decidable?