Cofinality $\operatorname{cf}(\alpha)$, and regular and singular cardinals

Let $\alpha$ be an ordinal (Ordinal (von Neumann)). The cofinality of $\alpha$ is

$\operatorname{cf}(\alpha) \;:=\; \text{the least ordinal } \beta \text{ for which some } f : \beta \to \alpha \text{ has cofinal range},$

cofinal range meaning that $f[\beta]$ is a cofinal subset of $\alpha$ (Cofinal subset of an ordinal): every $\zeta \in \alpha$ satisfies $\zeta \le f(\xi)$ for some $\xi \in \beta$. That such a least ordinal exists, and that a witnessing map of that length may be taken strictly increasing, is For every ordinal $\alpha$ there is a least ordinal $\beta$ admitting a map $\beta \to \alpha$ with cofinal range, and that map may always be taken strictly increasing, and both are theorems of ZF. So $\operatorname{cf}$ is defined at every ordinal, without any choice principle.

Regular and singular. An infinite cardinal $\kappa$ — a cardinal (Cardinal (initial ordinal) and cardinality) with $\omega \subseteq \kappa$ (Cardinal sum $\kappa \oplus \lambda$, product $\kappa \otimes \lambda$ and exponentiation $\kappa^{\lambda}$, and why they are written apart from the ordinal operations), for instance any $\aleph_\alpha$ (The successor cardinal $\kappa^{+}$, the alephs $\aleph_\alpha$, the beths $\beth_\alpha$, successor and limit cardinals, and the identifications $\aleph_0 = \omega$ and $\aleph_1 = \omega_1$) — is

• regular when $\operatorname{cf}(\kappa) = \kappa$;
• singular when $\operatorname{cf}(\kappa) \ne \kappa$.

The two cases are exhaustive by definition, and by $\operatorname{cf}(\alpha) \le \alpha$; $\operatorname{cf}(0) = 0$ and $\operatorname{cf}(\alpha + 1) = 1$; for a limit ordinal $\lambda$ the value $\operatorname{cf}(\lambda)$ is an infinite cardinal with $\operatorname{cf}(\operatorname{cf}(\lambda)) = \operatorname{cf}(\lambda)$, so it is regular; and every cofinal subset of $\lambda$ has cardinality at least $\operatorname{cf}(\lambda)$, a value that is attained ↗ singular means exactly $\operatorname{cf}(\kappa) < \kappa$, since $\operatorname{cf}(\alpha) \le \alpha$ always holds.

Why regularity is defined for cardinals and not for ordinals. The definition of $\operatorname{cf}$ applies to every ordinal, and it must, because the construction quantifies over maps into $\alpha$ of every length. But $\operatorname{cf}(\alpha) = \alpha$ is an uninteresting condition on a general ordinal: it fails at $\omega + 1$ and at $\omega \cdot 2$ for reasons that have nothing to do with size, and it holds only at $0$, at $1$, and at infinite cardinals, where it is exactly the regularity defined above and so fails at every singular one. Calling an ordinal regular would therefore say nothing new, which is why the words are attached to cardinals here.

What a singular cardinal is, in one sentence. A cardinal that is reachable from below by fewer than $\kappa$ steps: there is a strictly increasing family of ordinals below $\kappa$, indexed by an ordinal strictly shorter than $\kappa$, whose supremum is $\kappa$. That is exactly the failure of regularity, and $\aleph_0$ is regular in ZF; assuming the Axiom of Choice every successor aleph $\aleph_{\alpha+1}$ is regular; $\operatorname{cf}(\aleph_\omega) = \aleph_0$, so $\aleph_\omega$ is singular, and under choice it is the least singular infinite cardinal exhibits a cardinal for which it happens.

Why $\operatorname{cf}(\kappa)$ being a regular cardinal is a theorem and not part of the definition. Regularity is defined through $\operatorname{cf}$, so building "$\operatorname{cf}(\kappa)$ is regular" into the definition would make the definition refer to itself. The statement is true, and it is $\operatorname{cf}(\alpha) \le \alpha$; $\operatorname{cf}(0) = 0$ and $\operatorname{cf}(\alpha + 1) = 1$; for a limit ordinal $\lambda$ the value $\operatorname{cf}(\lambda)$ is an infinite cardinal with $\operatorname{cf}(\operatorname{cf}(\lambda)) = \operatorname{cf}(\lambda)$, so it is regular; and every cofinal subset of $\lambda$ has cardinality at least $\operatorname{cf}(\lambda)$, a value that is attained ↗; it is recorded here as the item that discharges the naming obligation of this definition, and nothing above depends on it.

Only one notion of "cofinal" exists in this library. Cofinal subset of an ordinal introduces cofinal subsets, because the boundedness theorem for $\omega_1$ needs them, and deliberately introduces neither the cofinality function nor the regular/singular vocabulary. Both are introduced here, and the definition above is written in exactly that item's terms, so no second notion is created.