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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
"There are good reasons why the theorems should all be easy and the definitions hard." --Michael Spivak, Preface to "Calculus on Manifolds", Addison-Wesley: 1965.
This may be a little off theme for …