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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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Intersection of integers and rationals defined by logic
Consider the class of sets of the form $X \cap Y$ where $X \subseteq \mathbb{N}^d$ is defined in FO($\mathbb{N}, +$) and $Y \subseteq \mathbb{Q}^d$ is defined in FO($\mathbb{Q}, +, \leq$). Clearly, th …