Cohen's Independence of the Continuum Hypothesis

Cohen proved in 1963 that the Continuum Hypothesis ( $2^{\aleph_0} = \aleph_1$ ) is independent of ZFC: it can neither be proved nor disproved from the standard axioms. Combined with Gödel's earlier consistency result, this completely resolves the status of CH relative to ZFC.

Statement

Theorem7.1Independence of CH

The Continuum Hypothesis is independent of ZFC:

(Gödel, 1940) If ZFC is consistent, then ZFC + CH is consistent. (Proved via $L$ .)
(Cohen, 1963) If ZFC is consistent, then ZFC + $\neg$ CH is consistent. (Proved via forcing.)

Proof

We sketch Cohen's forcing argument for (2). Start with a countable transitive model $M \models \text{ZFC} + \text{CH}$ . Use the forcing notion $\mathbb{P} = \mathrm{Fn}(\omega_2^M \times \omega, 2)$ : finite partial functions from $\omega_2^M \times \omega$ to $\{0,1\}$ , ordered by reverse inclusion.

A generic filter $G$ determines $\omega_2^M$ many new subsets of $\omega$ (one for each $\alpha < \omega_2^M$ ). Key properties:

$\mathbb{P}$ is c.c.c. (by a $\Delta$ -system argument), so $\mathbb{P}$ preserves all cardinals.
New reals are distinct: For $\alpha \neq \beta$ , the generically added functions $g_\alpha, g_\beta: \omega \to \{0,1\}$ differ.
No new cardinals collapsed: Since $\mathbb{P}$ is c.c.c., $\omega_1^M = \omega_1^{M[G]}$ and $\omega_2^M = \omega_2^{M[G]}$ .

Therefore in $M[G]$ : $2^{\aleph_0} \geq \aleph_2$ , so CH fails. $\square$

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RemarkEaston's Theorem

Cohen's method extends dramatically: Easton showed that the function $\kappa \mapsto 2^\kappa$ on regular cardinals can be essentially arbitrary (subject to König's theorem: $\operatorname{cf}(2^\kappa) > \kappa$ ). However, the behavior of $2^\kappa$ for singular $\kappa$ is much more constrained (Shelah's pcf theory).

ExampleMany Models of Set Theory

By varying the forcing notion, one can construct models where $2^{\aleph_0}$ equals $\aleph_1$ , $\aleph_2$ , $\aleph_{17}$ , $\aleph_{\omega+1}$ , or any regular cardinal satisfying König's constraint. This demonstrates the enormous flexibility of ZFC: it leaves the size of the continuum almost completely undetermined.