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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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The best text to study both incompleteness theorems
If you can read Spanish, an excellent text which both formally proves and philosophically treats both Incompleteness theorems is Carlos Ivorra Castillo's "Lógica y teoría de conjuntos" which is freely …