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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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Proof of second incompleteness theorem for Set theory without Arithmetization of Syntax
As mentioned in the comments, there are semantic proofs in set theory (thus non constructive, i.e they do not provide a recipe for constructing an undecidable sentence). However, Goedel himself first …