Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice

Over ZF, the generalised continuum hypothesis implies the Axiom of Choice.

Precisely: assume ZF, and assume GCH in the choice-free form "for every infinite set $A$ there is no set $B$ with $A\prec B\prec \mathcal{P}(A)$", where $X\prec Y$ means $X$ injects into $Y$ but not conversely. Then every set can be well-ordered, and so the Axiom of Choice holds.

The argument (Lindenbaum and Tarski announced it in 1926; Sierpiński gave the published proof in 1947) runs through Hartogs numbers. For a set $A$ let $\mathrm{\aleph }(A)$ be the least ordinal not injecting into $A$, which exists in ZF. One shows in ZF that $\mathrm{\aleph }(A)⪯\mathcal{P}(\mathcal{P}(\mathcal{P}(A)))$, and then uses GCH three times, on $A$, on $\mathcal{P}(A)$ and on $\mathcal{P}(\mathcal{P}(A))$, to force $A$ into bijection with an ordinal.

• Not proved in this library. Choice-free cardinal arithmetic is now developed elsewhere in the library — A set equinumerous with some ordinal has a least such ordinal, that ordinal is a cardinal, and equinumerous sets get the same one; no choice principle is used ↗, Hessenberg's theorem and the successor-cardinal lemma between them give $A+A$ and the successor-cardinal step for a well-orderable $A$ — but that is not enough by itself: the argument here needs the comparison of $\mathrm{\aleph }(A)$ against iterated power sets of an arbitrary, not-yet-well-ordered $A$, which is a genuinely different and harder computation, and remains undeveloped.

• What would prove it. The choice-free cardinal arithmetic the library now has, on top of the Hartogs construction, gets partway there. The ingredients the library already has are Hartogs: an ordinal that does not inject into a given set ↗ (for every set there is a least ordinal that does not inject into it, proved with no choice at all) and, since this build, A set equinumerous with some ordinal has a least such ordinal, that ordinal is a cardinal, and equinumerous sets get the same one; no choice principle is used ↗, Hessenberg's theorem and the successor-cardinal lemma. What is still missing is the comparison of $\mathrm{\aleph }(A)$ with $\mathcal{P}(\mathcal{P}(A))$ and $\mathcal{P}(\mathcal{P}(\mathcal{P}(A)))$ for an arbitrary infinite $A$, which is not in this library.

• Why it matters here. It settles the status of GCH as a hypothesis: GCH is not a harmless size assumption to be added to ZF, it is at least as strong as the Axiom of Choice (The Axiom of Choice ↗). This is why The continuum hypothesis and its generalisation are independent of ZFC ‡ is stated over ZFC rather than ZF, and why The continuum hypothesis, and what this page does not prove ↗ records that GCH implies CH and reaches far beyond it. What carries the argument is the generalised form: the proof applies the hypothesis to $A$, to $\mathcal{P}(A)$ and to $\mathcal{P}(\mathcal{P}(A))$ for an arbitrary infinite $A$, and the single instance $A=\mathbb{N}$ that is CH gives none of that. This library records no result at all about whether CH alone implies the Axiom of Choice, in either direction, and no page here may assert one.

• Conditional discipline. This one statement is not an independence result and needs no consistency hypothesis: it is an ordinary implication provable in ZF. It appears on this page only because its proof, not its status, is beyond the library's current machinery.