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first-order and higher-order logic, model theory, set theory, proof theory, computability theory, formal languages, definability, interplay of syntax and semantics, constructive logic, intuitionism, philosophical logic, modal logic, completeness, Gödel incompleteness, decidability, undecidability, theories of truth, truth revision, consistency.
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Is any true sentence in the second-order Peano Axioms provable
Between them, the previous comments and replies have already covered all of the relevant points, but here in any case is a direct, unsophisticated answer:
Any effective procedure (i.e. anything you c …