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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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Extending the class of primitive recursive functions with higher order recursion schema
a constructive (from intuitionistic point of view) of higher order calculus of functionals and relations is at
https://arxiv.org/abs/1501.03043