Worked solution: Gödel and Cohen: the continuum hypothesis is independent of ZFC (1963)

Step 2 of 9: What "independent of

\mathrm{ZFC}

" actually means

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In plain words

For two thousand years, mathematicians tried to prove Euclid's fifth postulate (about parallel lines) from his other four axioms, and always failed; eventually it turned out the postulate is genuinely optional — geometries where it holds (flat, Euclidean) and geometries where it fails (curved, non-Euclidean) are both perfectly consistent.

$\mathrm{ZFC}$ (Zermelo–Fraenkel set theory with choice) is the standard rulebook mathematicians use for building sets. Saying $\mathrm{CH}$ is independent of $\mathrm{ZFC}$ means it behaves exactly like Euclid's fifth postulate: the rulebook itself simply does not decide whether $\mathrm{CH}$ or its opposite $\neg\mathrm{CH}$ holds, and both choices lead to equally consistent worlds of sets.

\mathrm{ZFC} \nvdash \mathrm{CH} \quad\text{and}\quad \mathrm{ZFC} \nvdash \neg\mathrm{CH}

Detailed analysis

A statement $\varphi$ is independent of a consistent theory $T$ if neither $\varphi$ nor $\neg\varphi$ can be proved from $T$ 's axioms; equivalently, both $T + \varphi$ and $T + \neg\varphi$ are themselves consistent theories. Proving independence therefore always takes two separate halves, each a relative consistency result.

Half one: build a model (a universe of sets) in which every $\mathrm{ZFC}$ axiom holds and $\mathrm{CH}$ also holds, showing $\mathrm{ZFC} \nvdash \neg\mathrm{CH}$ . Half two: build a different model in which every $\mathrm{ZFC}$ axiom holds but $\neg\mathrm{CH}$ holds instead, showing $\mathrm{ZFC} \nvdash \mathrm{CH}$ . Neither half alone settles independence — Gödel supplying only half one by 1940 left open the possibility that $\mathrm{CH}$ might still be provable from $\mathrm{ZFC}$ ; only Cohen's 1963 construction of the second model closed that gap for good.

The next three steps present Gödel's half (Steps 3–4), then the following three present Cohen's half (Steps 5–7), before Step 8 combines them into the full independence theorem.

Terms in this step

Model of a theory: A structure (here, a universe of sets together with the membership relation $\in$ ) in which every axiom of a given theory comes out true. Building a model of $\mathrm{ZFC} + \varphi$ is how one proves $\varphi$ is consistent with $\mathrm{ZFC}$ .

Knowledge used in this step

The ZFC axioms