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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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Proper classes and their consequences
(Answer for 2.) One reason most mathematicians can avoid the theory of proper classes is through the use of Grothendieck universes. In brief, one chooses a strongly inaccessible cardinal $\kappa$, wh …