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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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What do models where the CH is false look like?
You must start slightly further back. The Axiom of Choice (AC) and the Axiom of Determinacy (AD) are both very desirable, but contradict one another. We usually feel AC has more claim to truth since …