The Feferman-Levy model: the reals as a countable union of countable sets

If ZF is consistent, then ZF is consistent with all of the following holding simultaneously:

• $\mathbb{R}$ is a countable union of countable sets;
• the first uncountable ordinal ${\omega }_1$ is singular, indeed $\mathrm{c}\mathrm{f}({\omega }_1)=\omega$;
• consequently "a countable union of countable sets is countable" fails, and ${\omega }_1$ is not regular.

Feferman and Levy (1963) obtain this by collapsing: starting from a model of ZFC, force with the finite-support product that makes each ${\mathrm{\aleph }}_n$ of the ground model countable, for $n\in \mathbb{N}$, and take the symmetric submodel with finite supports. The ground model's ${\mathrm{\aleph }}_{\omega }$ becomes the new ${\omega }_1$, and it is the supremum of the countably many ordinals ${\mathrm{\aleph }}_n$, each now countable, so its cofinality is $\omega$. The reals of the extension are the union over $n$ of the reals added at stage $n$, and each of those layers is countable in the extension.

Note what does not fail: ${\omega }_1$ still exists, and $\mathbb{R}$ is still uncountable. A countable union of countable sets is being exhibited whose union is uncountable, which is possible exactly because no enumeration of the layers can be chosen uniformly.

• Not proved in this library. The Levy collapse and the symmetric submodel are not developed here.

• What would prove it. The forcing track of Cohen 1963: ZF does not prove the Axiom of Choice ‡, specialised to the Levy collapse $\mathrm{C}\mathrm{o}\mathrm{l}\mathrm{l}(\omega ,<{\mathrm{\aleph }}_{\omega })$ with finite supports, plus the cofinality computation in the symmetric model. Ordinal and cardinal arithmetic beyond Cardinal (initial ordinal) and cardinality ↗ is also needed to state the cofinality claim properly.

• Why it matters later. The countable union theorem Countable unions of at most countable sets, assuming ${\mathrm{A}\mathrm{C}}_{\omega }$ ↗ is proved here from The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$) ↗. Once this model is constructed locally, it will show that the hypothesis cannot simply be dropped. It is also the standing warning attached to ${\omega }_1$: any counterexample built on the ordinal space $[0,{\omega }_1)$ and its sequential compactness is silently spending a choice principle, because in this model ${\omega }_1$ has countable cofinality and those arguments collapse. Note also that $\mathbb{R}$ is uncountable (Cantor's nested intervals, 1874) ↗ survives untouched: it is a theorem of ZF, and it is not in tension with $\mathbb{R}$ being a countable union of countable sets.

• Conditional discipline. Everything above is relative to the consistency of ZF, and this library never asserts that the reals are a countable union of countable sets, only that ZF alone cannot refute it.