Blass 1977: a model of ZF with no free ultrafilter on any set

If ZF is consistent, then ZF is consistent with the statement that every ultrafilter on every set is principal. Equivalently: there is a model of ZF containing no free ultrafilter at all, on any set whatsoever.

This is Blass (1977). As a statement it strengthens Feferman 1965: ZF does not prove that a free ultrafilter on the naturals exists ‡, where the conclusion was obtained only for $\mathbb{N}$: "every ultrafilter on every set is principal" implies "every ultrafilter on $\mathbb{N}$ is principal", so the consistency of the first is the stronger result. The model is again obtained by forcing and symmetry, but the construction is not reproduced here: the published note is a summary (see the citation remark below), and this library has not read a full account of it, so it records the conclusion and not the method.

• Not proved in this library. No forcing or symmetric-model machinery is developed here.

• What would prove it. The forcing and symmetric-model machinery named in Cohen 1963: ZF does not prove the Axiom of Choice ‡, with the additional uniformity argument that handles arbitrary sets rather than a single fixed set.

• Why it matters here. FALSE, once the ultrafilter lemma is available: every ultrafilter is principal ↗ refutes the claim that every ultrafilter is principal, and does so only after invoking The ultrafilter lemma, from the Axiom of Choice: every filter extends to an ultrafilter ↗, which costs a choice principle. This item is the sharp statement of why that cost is unavoidable in the strongest possible sense: it is not that the refutation happens to use choice for the particular filter of tails on $\mathbb{N}$, it is that in ZF alone one cannot get a single non-principal instance of Ultrafilter ↗ anywhere.

• On the citation. The primary source is a short note in the Bulletin de l'Académie Polonaise des Sciences (volume 25, number 4, pages 329-331), which has no open electronic edition and which the reviewing literature describes as a summary rather than a full account. The reference url above therefore points at the standard survey statement of the fact, "ZF alone does not even imply that there exists a non-principal ultrafilter on some set", rather than at the note itself; the note's full bibliographic details are given in the reference title. The survey page states the fact but does not name Blass, so the attribution rests on the bibliographic record, not on that page.

• Conditional discipline. Relative to the consistency of ZF, as always. The claim is that ZF cannot refute "every ultrafilter is principal", not that the statement is true.