# Continuum Hypothesis is Independent of ZFC

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## Theorem

The Continuum Hypothesis can be neither proved nor disproved from the axioms of either Zermelo-Fraenkel set theory (ZF) or ZFC.

## Proof

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## Historical Note

Between $1938$ and $1940$ (accounts differ), Kurt Gödel showed that the **Continuum Hypothesis (CH)** is formally consistent with the axioms of Zermelo-Fraenkel set theory (ZF) with or without the Axiom of Choice (ZFC).

Between $1963$ and $1966$, Paul Cohen showed that the negation of both the **Continuum Hypothesis** and the **Generalized Continuum Hypothesis** are also consistent with the axioms of ZF and ZFC.

Thus, neither ZF and ZFC are strong enough to settle the question of the **Continuum Hypothesis** either way.

## 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