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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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Finite presentability and elementary equivalence
The problem is known and still open but does not seem hopeless. I would start with considering Zilber's old example of two f.g. non-isomorphic nilpotent of class $2$ groups which are e.e.: B.I. Zilʹbe …