Continuum hypothesis

Independent of the axiomsFoundations of mathematicsHilbert #1

Statement

There is no set whose cardinality is strictly between that of the integers $\mathbb{Z}$ and that of the real numbers $\mathbb{R}$ ; equivalently $2^{\aleph_0} = \aleph_1$ .

Gödel constructed the constructible universe $L$ in 1938–40 to show CH is consistent with ZFC; Cohen invented forcing in 1963 to build models of ZFC + ¬CH, showing CH is also independent in the other direction. Together their work settles that CH cannot be decided from ZFC.

Posed by, 1878Georg Cantor Solved by, 1963Kurt Gödel Solved by, 1963Paul Cohen

Cohen's forcing technique became the standard tool for independence proofs throughout set theory. Since 1963, set theorists have explored additional axioms (such as Martin's Axiom, large cardinals, or Woodin's $\Omega$ -logic programme) that might settle CH in a philosophically motivated way, but no such axiom has achieved consensus.

References

Georg Cantor (1878). Ein Beitrag zur Mannigfaltigkeitslehre
Kurt Gödel (1940). The Consistency of the Continuum Hypothesis · DOI:10.1515/9781400881635
Paul J. Cohen (1963). The Independence of the Continuum Hypothesis