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 $\mathrm{cf}(\omega_1) = \omega$;
• 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 $\aleph_n$ of the ground model countable, for $n \in \mathbb{N}$, and take the symmetric submodel with finite supports. The ground model's $\aleph_\omega$ becomes the new $\omega_1$, and it is the supremum of the countably many ordinals $\aleph_n$, 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 $\mathrm{Coll}(\omega, {<}\aleph_\omega)$ 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 here. Two library results lean on it directly. It is the external fact behind FALSE: countable unions of countable sets are countable is a theorem of ZF ↗: the countable union theorem Countable unions of at most countable sets, assuming $\mathrm{AC}_\omega$ ↗ is proved here from The Axiom of Countable Choice ($\mathrm{AC}_\omega$) ↗, and this model shows the hypothesis cannot 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.