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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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Taking a theorem as a definition and proving the original definition as a theorem
An example which is not quite what is being asked, and which is certainly much less sophisticated than the other answers, but which comes up in my calculus teaching around this week each year: continu …