Clarifying ordinals

TL;DR

This paper explores the limits of computability of isomorphisms between copies of certain ordinals using forcing and set-theoretic assumptions, revealing deep connections between logic, set theory, and computability.

Contribution

It demonstrates that for many ordinals, isomorphisms are not computable in the join of certain descriptive set-theoretic sets, under various set-theoretic assumptions.

Findings

Existence of non-computable isomorphisms for ordinals in a club subset of ω₁.

Under V=L, this non-computability result is nearly optimal.

Assuming large cardinals, similar non-computability results extend to all projective functions.

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 $\mathsf{V=L}$, this is close to optimal; on the other hand, assuming large cardinals the same (and more) holds for every projective functional.