Clarifying ordinals

Abstract

We use forcing over admissible sets to show that, for every ordinal $\alpha$ in a club $C\subset {\omega }_1$, there are copies of $\alpha$ such that the isomorphism between them is not computable in the join of the complete ${\Pi }_1^1$ set relative to each copy separately. Assuming $\mathsf{V}=\mathsf{L}$, this is close to optimal; on the other hand, assuming large cardinals the same (and more) holds for every projective functional.