# 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