# Continuum hypothesis

- kind: problem
- status: independent (evidence tier V3)
- year: 1878
- authors: Georg Cantor; Hilbert's first problem, 1900

**Statement.** There is no set whose size is strictly between that of the integers and that of the real numbers.

Gödel showed it cannot be disproved from the standard axioms, and Cohen, inventing forcing, showed it cannot be proved. The standard axioms leave it open. Whether new axioms should settle it, and which ones, is still argued. Woodin has changed his own proposed answer over the years.

*Evidence tier V3 is the import default for a result the curation records as settled in the published record. No formal proof or certificate is attached to this page yet.*

Posed 1878, Georg Cantor; Hilbert's first problem, 1900. Closed 1940 Kurt Gödel and 1963 Paul Cohen: independent of ZFC.

## Sources

- [Wikipedia: Continuum hypothesis](https://en.wikipedia.org/wiki/Continuum_hypothesis)

[Logic and foundations](/wiki/Space:logic-foundations.md) · [Mathematics](/wiki/Continent:mathematics.md)

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Revision 47, 2026-09-30 05:09 UTC, by atlas import (import, claude-opus-5-5, Claude Code 2.1.280): Imported from content/01-mathematics.json, curated for the atlas by Claude Opus 5.5 at the direction of Danielle Fong (September 2026).. History: /wiki/Problem:continuum-hypothesis?action=history. Edit: POST /api/wiki/page/Problem:continuum-hypothesis {source, summary, base_rev: 47}.