The continuum hypothesis and its generalisation are independent of ZFC

Write CH for the continuum hypothesis, the assertion that no set $A$ satisfies $\mathbb{N}\prec A\prec \mathcal{P}(\mathbb{N})$, where $X\prec Y$ means that $X$ injects into $Y$ but $Y$ does not inject into $X$. Write GCH for the generalised continuum hypothesis, the assertion that for no infinite $A$ is there a $B$ with $A\prec B\prec \mathcal{P}(A)$.

If ZFC is consistent, then all four of ZFC + CH, ZFC + (not CH), ZFC + GCH and ZFC + (not GCH) are consistent. In particular neither CH nor GCH is decided by ZFC, unless ZFC is inconsistent.

The two halves come from the two constructions already recorded here. The constructible universe of Gödel 1938: ZF does not refute the Axiom of Choice ‡ satisfies GCH, and hence CH, which rules out a ZFC refutation of either. Cohen's forcing, the method of Cohen 1963: ZF does not prove the Axiom of Choice ‡, adjoins ${\mathrm{\aleph }}_2$ many generic subsets of $\mathbb{N}$ to a model of ZFC + GCH while preserving cardinals, producing a model of ZFC in which CH, and therefore GCH, fails.

In cardinal notation, using the alephs and beths of the aleph and beth hierarchies developed later in the library, GCH is the statement $2^{{\mathrm{\aleph }}_{\alpha }}={\mathrm{\aleph }}_{\alpha +1}$ for every ordinal $\alpha$, and CH is its instance $\alpha =0$.

• Not proved in this library. Neither half is proved here. The cardinal arithmetic in which the statement is phrased above is now fully available — Cardinal (initial ordinal) and cardinality ↗, cardinal exponentiation and the aleph and beth hierarchies are all developed on a later page — what is absent is the forcing and inner-model machinery that would prove the independence statement itself.

• What would prove it. For consistency of CH, the inner model $L$ together with its condensation lemma. For consistency of the negation, the forcing $\mathrm{F}\mathrm{n}({\mathrm{\aleph }}_2\times \omega ,2)$ of finite partial functions, plus the countable chain condition and the $\mathrm{\Delta }$-system lemma, which are what show that this forcing collapses no cardinal. Both belong to a forcing and inner-model track that this library does not contain.

• Why it matters here. The continuum hypothesis, and what this page does not prove ↗ states CH immediately after Cantor's theorem: $A\prec \mathcal{P}(A)$ ↗ establishes the gap $\mathbb{N}\prec \mathcal{P}(\mathbb{N})$, and has to record that the obvious next question is not answerable. This item is the result it quotes. Note the gap that The continuum hypothesis, and what this page does not prove ↗ itself points out: the form of CH about $\mathbb{R}$ rather than about $\mathcal{P}(\mathbb{N})$ needs $\mathbb{R}\approx \mathcal{P}(\mathbb{N})$, which the library now proves in ZF, on a later page, so the two forms are interchangeable after all — the gap The continuum hypothesis, and what this page does not prove ↗ records was closed by a later page, not a standing limitation of this remark's own claim.

• Conditional discipline. Every clause above is relative to the consistency of ZFC. "CH is independent of ZFC" is shorthand and is never used here as a standalone assertion.