Axiom of Choice is Independent of ZF

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Theorem

The Axiom of Choice can be neither proved nor disproved from the axioms of Zermelo-Fraenkel set theory.


Proof


Historical Note

Kurt Gödel showed that the Axiom of Choice is not disprovable in Zermelo-Fraenkel set theory.

Paul Cohen showed that neither is the Axiom of Choice provable in Zermelo-Fraenkel set theory.


Sources