Gitik 1980: consistently, every uncountable cardinal is singular

If ZFC together with "there is a proper class of strongly compact cardinals" is consistent, then so is ZF + "every uncountable cardinal is singular".

That is: relative to that large-cardinal hypothesis, there is a model of ZF in which no uncountable cardinal is regular at all, so ${\mathrm{\aleph }}_1$, ${\mathrm{\aleph }}_2$ and every other uncountable aleph has cofinality $\omega$. Gitik (1980) obtains it by an iterated Prikry-style forcing over a model with a proper class of strongly compact cardinals, followed by a symmetric submodel.

• Not proved in this library. No forcing, no large cardinals, and no symmetric submodels: that part of the machinery is entirely outside the library's stack. The cofinality function $\mathrm{cf}(\alpha )$ and the words regular and singular are themselves defined here (Cofinality $\mathrm{cf}(\alpha )$, and regular and singular cardinals ↗), and the library proves one regularity fact under choice (${\mathrm{\aleph }}_0$ is regular in ZF; assuming the Axiom of Choice every successor aleph ${\mathrm{\aleph }}_{\alpha +1}$ is regular; $\mathrm{cf}({\mathrm{\aleph }}_{\omega })={\mathrm{\aleph }}_0$, so ${\mathrm{\aleph }}_{\omega }$ is singular, and under choice it is the least singular infinite cardinal ↗) — but that is choice-theoretic ground truth, the opposite of what Gitik's result denies, and nothing here touches the choice-free question this remark records.

• What would prove it. Large-cardinal theory (strongly compact cardinals and their embeddings), Prikry forcing and its iterations, and symmetric extensions. This is the deepest result recorded on this page and is far outside the library's stack.

• Why it matters here. It is the extreme point of the phenomenon that The Feferman-Levy model: the reals as a countable union of countable sets ‡ introduces. That model makes ${\omega }_1$ singular; Gitik's makes every uncountable cardinal singular. The practical reading for this library is that regularity of an uncountable cardinal is never a free structural fact about Cardinal (initial ordinal) and cardinality ↗: it is a consequence of choice, and without choice it can fail everywhere at once. What survives in ZF is the much weaker existence statement Hartogs: an ordinal that does not inject into a given set ↗, which is why the library leans on Hartogs numbers, and why the one regularity statement it does prove — that no at most countable subset of ${\omega }_1$ is cofinal in it — carries the Axiom of Countable Choice as an explicit standing hypothesis rather than reading it off the structure. The proved choice ledger: hypotheses, equivalences, and upper bounds ↗ records that distinction.

• Conditional discipline. The hypothesis is a large-cardinal consistency assumption strictly stronger than Con(ZFC), and the conclusion is a relative consistency statement. Nothing here asserts that strongly compact cardinals exist.