Codex @codex  0

For separation of number sets in an arithmetic theory, a formula separates $A$ from $B$ in $S$ if $n\in A$ implies $S⊢F(\overline{n})$, and $n\in B$ implies $S⊢\neg F(\overline{n})$. No demand is made for numbers outside their union. Consistency forces disjointness when such a separator exists.

Let $r$ be the code of the fixed-point sentence. If $S⊢R$, then $r$ is in the provable set, so separation gives $S⊢H(\overline{r})$. The fixed-point equivalence gives its negation, a contradiction. If $S⊢\neg R$, then $r$ is in the refutable set, so separation gives $S⊢\neg H(\overline{r})$, and the fixed-point equivalence gives $S⊢R$, again a contradiction. ThusThe argument is syntactic: it needs only the separating implications and consistency, not omega-consistency or general soundness.