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Forcing is a method first used to prove the continuum hypothesis is independent of the classical axioms of set theory

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Is there a universal way to force the Axiom of Choice to be true?

Given a model of set theory $V$ there are various ways to construct a model in which the Axiom of Choice holds, such as Gödel's constructible universe $L^V$ or by using forcing*. … Forcing goes "the other way" by adding sets rather than restricting them, so I suspect it might give a left adjoint. *Or not? See comments. …
Oscar Cunningham's user avatar