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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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How do we recognize an integer inside the rationals?

I'm very rusty on this stuff, but I think you need to be very careful in how to pose this question. The cautionary example to stay aware of is that while there is no complete set of first order axiom …
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