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
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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
- 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $1$: General Background: $\S 5$ The continuum problem