# Continuum Hypothesis

**Type:** Theorem  
**Domain:** Foundations & Logic  
**Posed:** 1878, Georg Cantor  
**Status:** INDEPENDENT of ZFC (Gödel 1940 + Cohen 1963)  
**Codex URL:** /mathematics/foundations-logic/continuum-hypothesis/  
**Entry status:** Live — v1.0 (2026-05-27)

## Summary
Is there a cardinality strictly between the integers and the reals? Proven independent of ZFC.

## Cantor's diagonal argument (1891)
Proves reals cannot be put in one-to-one correspondence with integers — reals are a strictly larger infinity.

## CH stated formally
2^ℵ₀ = ℵ₁ — continuum is the immediate next cardinal after the integers.

## Two-part resolution
Gödel (1940): CH is consistent with ZFC (constructible universe L). Cohen (1963): ¬CH is also consistent with ZFC (forcing technique) — together proving CH is independent. Cohen won the 1966 Fields Medal — the only one awarded primarily for mathematical logic.

## Hilbert's Problem 1
CH was the very first item on Hilbert's 1900 list of 23 problems.

## Generalized Continuum Hypothesis (GCH)
Extends CH to every level of the cardinal hierarchy: 2^(ℵₙ)=ℵₙ₊₁ for all n. Also independent of ZFC.

## Sources
### Tier 1
- Cohen, P.J. (1963). PNAS.
- Gödel, K. (1940). Princeton University Press.
### Tier 2
- Cohen, P.J. (1966). Set Theory and the Continuum Hypothesis.

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*Mathematics Codex entry v1.0 — added 2026-05-27 — thecodex.expert/mathematics/*