GCH: for every ordinal α, there is no cardinal strictly between ℵ_α and \(2^{ℵ_α}\). Unlike CH, which is independent of ZFC (Gödel 1940 + Cohen 1963), GCH fails in many forcing extensions where CH holds, and its status under large-cardinal axioms is more nuanced.
Sierpiński championed GCH as the natural generalisation. Easton's theorem (1970) showed that regular-cardinal values of the continuum function are almost completely unconstrained subject to monotonicity — meaning GCH can fail spectacularly at regular cardinals while holding elsewhere. Singular cardinals behave differently: Silver's theorem constrains failures at singular cardinals of uncountable cofinality, and the Singular Cardinal Hypothesis (SCH) is deeply tied to large cardinals. Whether GCH holds 'generically' given sufficient large-cardinal assumptions is one of set theory's deepest live questions.
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