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