generalized continuum hypothesis

mathematics

Also known as: GCH

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June 29, 2026

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The generalized continuum hypothesis (GCH) is a statement in set theory that extends Georg Cantor's original continuum hypothesis. The continuum hypothesis proposes that there is no set with a cardinality (size of a set) strictly between that of the natural numbers ((\aleph_0)) and the real numbers ((c)). The GCH generalizes this by stating that for any ordinal number (\alpha), the cardinality of the power set of a set with cardinality (\aleph_\alpha) is equal to the next cardinal number, (\aleph_{\alpha+1}).12

Mathematicians Kurt Gödel and Paul Cohen proved that the GCH (and the original continuum hypothesis) is independent of the standard axioms of set theory (Zermelo-Fraenkel axioms). This means that these axioms neither prove nor disprove the GCH, implying that there may be multiple consistent set theories with different properties regarding the continuum hypothesis.134

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set theory
Present status of axiomatic set theory
… (CH) and its extension, the generalized continuum hypothesis (GCH), are also of profound importance. In the following discussion of these questions, ZF denotes Zermelo-Fraenkel set theory without AC. The first finding was obtained by Kurt Gödel in 1939. He proved that AC and GCH are consistent relative to ZF (i.e.,…
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… (CH) and its extension, the generalized continuum hypothesis (GCH), are also of profound importance. In the following discussion of these questions, ZF denotes Zermelo-Fraenkel set theory without AC. The first finding was obtained by Kurt Gödel in 1939. He proved that AC and GCH are consistent relative to ZF (i.e.,…
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continuum hypothesis
A stronger statement is the generalized continuum hypothesis (GCH): 2^ℵ_α = ℵ_α + 1 for each ordinal number α. The Polish mathematician Wacław Sierpiński proved that with GCH one can derive the axiom of choice.
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A stronger statement is the generalized continuum hypothesis (GCH): 2^ℵ_α = ℵ_α + 1 for each ordinal number α. The Polish mathematician Wacław Sierpiński proved that with GCH one can derive the axiom of choice.
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metalogic
Generalizations and extensions of the Löwenheim-Skolem theorem
…established only when the “generalized continuum hypothesis” is assumed, according to which the next highest cardinality for an infinite set is that of its power set.)
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…established only when the “generalized continuum hypothesis” is assumed, according to which the next highest cardinality for an infinite set is that of its power set.)
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