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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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Finite versions of Godel' s incompleteness
It is important to include the axioms as well, since there is a theorem of logic that an axiom system can be changed equivalently so that any given proof can be made arbitrarily sorter, or on the cont …