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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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Essential reads in the philosophy of mathematics and set theory
I think since the fundamentals of the philosophy of mathematics have been well-established above, I'd like to submit Albert Lautmann's Mathematics, Ideas and the Physical Real and the more recent (and …