We use forcing over admissible sets to show that, for every ordinal α in a club C⊂ω₁, there are copies of α such that the isomorphism between them is not computable in the join of the complete Π¹₁ set relative to each copy separately. Assuming V=L, this is close to optimal; on the other hand, assuming large cardinals the same (and more) holds for every projective functional.
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Noah Schweber (2024) studied this question.
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