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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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Is there an analogue of the Lost Melody Theorem in ordinary recursion theory and if not, why...

It was already mentioned that the "domain enumeration" property of classical Turing machines ban them from having lost melodies right away. But (as was already hinted in the comments) when one introdu …
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