Exercise 2100Logic, sets and proofs › Countability and cardinalityM1Added 3 Nov 2026

The generalized continuum hypothesis (statement)

#GCH#generalized continuum#set theory

Statement

State the generalized continuum hypothesis and its relation to CH.

Hint

GCH: for every infinite cardinal $\kappa$ , $2^\kappa=\kappa^+$ (the next cardinal).

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GCH. $2^\kappa=\kappa^+$ for all infinite cardinals $\kappa$ .

Special case. $\kappa=\aleph_0$ : $2^{\aleph_0}=\aleph_1$ , which is CH.

Independence. GCH is also independent of ZFC (Gödel, Cohen).

Conclusion. GCH generalizes CH to all infinite cardinals.