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Second order logic with Henkin semantics is axiomatizable in first order logic, for example, using truth predicates and comprehension axioms for all arities. Is it possible to provide a finite axiomatization as well?

Akuri
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  • Do you mean finitely axiomatizable or recursively axiomatizable? – hasManyStupidQuestions Jan 03 '23 at 16:08
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    Finitely, similar to how NBG set theory can. – Akuri Jan 09 '23 at 04:35
  • I don't know, but this might touch on the differences and relationships between "finitely axiomatizable"and "recursively axiomatizable" (because the comprehension schemas for general semantics / Henkin semantics https://plato.stanford.edu/entries/logic-higher-order/#AxioSecoOrdeLogi seem to basically be the same as those in Peano arithmetic or ZFC) this answer to a related question might help? https://math.stackexchange.com/a/4613983/606791 https://math.stackexchange.com/questions/955523/finite-list-of-axioms-of-mathsfaca-0-reference That it's possible for ACA0 suggests the answer could be yes – hasManyStupidQuestions Jan 16 '23 at 16:38

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