Exercise 2099Logic, sets and proofs › Countability and cardinalityM1Added 3 Nov 2026

The continuum hypothesis (statement)

#continuum hypothesis#aleph#independence

Statement

State the continuum hypothesis and its independence from ZFC.

Hint

CH: $2^{\aleph_0}=\aleph_1$ , i.e. no cardinal strictly between $|\mathbb{N}|$ and $|\mathbb{R}|$ .

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CH. There is no set whose cardinality is strictly between $\aleph_0$ and $2^{\aleph_0}$ .

Independence (Gödel 1940, Cohen 1963). CH is independent of ZFC: neither CH nor its negation is provable from ZFC.

Conclusion. CH is undecidable in standard set theory.