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Sniffnoy comments on Edward Nelson claims proof of inconsistency in Peano Arithmetic - Less Wrong
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Strictly speaking, PA uses infinitely many axioms -- the induction axiom is actually an axiom schema, one axiom for each predicate you can plug into it. If you actually had it as one axiom quantifying over predicates, that would be second-order.
I think that Nelson denies that there is a completed infinity of predicates that you can plug into the schema.
Well, you'd certainly only need finitely many to prove inconsistency.