The continuum hypothesis and the later local consistency results

By Cantor's theorem: $A\prec \mathcal{P}(A)$ there is a strict gap $\mathbb{N}\prec \mathcal{P}(\mathbb{N})$ (Equinumerous sets, $A\approx B$ and $A⪯B$). In particular $\mathcal{P}(\mathbb{N})$ is uncountable (Finite, countably infinite, countable, uncountable), since a surjection $\mathbb{N}\to \mathcal{P}(\mathbb{N})$ would exist if it were at most countable (A nonempty set is at most countable iff it is a surjective image of $\mathbb{N}$) and the theorem forbids one; and so, by a completely different argument, is $\mathbb{R}$ ($\mathbb{R}$ is uncountable (Cantor's nested intervals, 1874)). The obvious next question is whether anything sits strictly in between.

The continuum hypothesis (CH) asserts that nothing does:

there is no set $A$ with $\mathbb{N}\prec A\prec \mathcal{P}(\mathbb{N})$.

Over ZFC this is equivalent to: every uncountable subset of $\mathcal{P}(\mathbb{N})$ is equinumerous with $\mathcal{P}(\mathbb{N})$ itself. The qualification matters. Passing from the displayed form to the subset form uses choice to well order an uncountable $A\subseteq \mathcal{P}(\mathbb{N})$ and obtain $\mathbb{N}\prec A$. Nothing here asserts that this passage is available in ZF; only the displayed form is used below. Determining the exact choiceless relationship between the formulations belongs to the later symmetric-model development.

Later local consistency results. Positive relative consistency of CH and GCH ↗ proves that $\mathrm{Con}(\mathrm{Z}\mathrm{F}\mathrm{C})$ implies consistency of both $\mathrm{Z}\mathrm{F}\mathrm{C}+\mathrm{C}\mathrm{H}$ and $\mathrm{Z}\mathrm{F}\mathrm{C}+\mathrm{G}\mathrm{C}\mathrm{H}$. Externally fixed-fragment relative consistency of not CH and not GCH ↗ proves, as an external metatheorem using a separate construction for each fixed finite fragment, the corresponding consistency implications for not CH and not GCH. Its contract claims no PA proof of a uniform refutation transformer. Together these imply that, if ZFC is consistent, it decides neither CH nor GCH: a proof of either sentence would contradict consistency of the extension by its negation.

What this page has not proved. CH is usually stated about $\mathbb{R}$: that every uncountable set of reals is equinumerous with $\mathbb{R}$. That form is equivalent to the one above only once one knows $\mathbb{R}\approx \mathcal{P}(\mathbb{N})$, which this library now proves, in ZF, on a later page. At this point in the reading order, though, the two uncountability results on this page are still genuinely separate facts: $\mathcal{P}(\mathbb{N})$ is uncountable by the diagonal argument, and $\mathbb{R}$ is uncountable by nested intervals, and the bridge between them is not available here — it needs binary expansions, which are developed much later, on the same later page. Nothing on this page depends on that bridge.

The countability results have their own proofs. Countability of $\mathbb{Q}$, uncountability of $\mathbb{R}$ and of the irrationals, and Cantor's theorem on this page are proved without a choice axiom. The later consistency comparisons are not premises of those arguments.

The generalised continuum hypothesis (GCH) asserts that there are no infinite $A$ and set $B$ with $A\prec B\prec \mathcal{P}(A)$. It implies CH by its instance at $A=\mathbb{N}$, since $\mathbb{N}$ is infinite by claim 4 of The pigeonhole principle on $\mathbb{N}$. The later Sierpiński: arbitrary-set GCH implies AC in ZF ↗ proves in ZF that this arbitrary-set formulation of GCH implies AC. This implication uses no consistency assumption and is distinct from the two consistency comparisons.