Cofinal subset of an ordinal

Let $\alpha$ be an ordinal (Ordinal (von Neumann)). A subset $C \subseteq \alpha$ is cofinal in $\alpha$, equivalently unbounded in $\alpha$, when

$\text{for every } \xi \in \alpha \text{ there is } \eta \in C \text{ with } \xi \le \eta.$

A subset that is not cofinal is bounded below $\alpha$: there is $\xi \in \alpha$ such that $\eta < \xi$ for every $\eta \in C$.

• At a limit ordinal, cofinal means the supremum is attained from below. If $\lambda$ is a limit ordinal (Successor and limit ordinals) and $C \subseteq \lambda$ is nonempty, then $C$ is cofinal in $\lambda$ if and only if $\sup C = \bigcup C = \lambda$ (claim (e) of Basic closure properties of ordinals). If $\bigcup C = \lambda$, then every $\xi \in \lambda$ lies in some $\eta \in C$, so $\xi < \eta$ and $C$ is cofinal. Conversely, if $C$ is cofinal then $\bigcup C \subseteq \lambda$, because each $\eta \in C$ satisfies $\eta \subseteq \lambda$ by transitivity; and for $\xi \in \lambda$ the ordinal $\xi^{+}$ again lies in $\lambda$ (Successor and limit ordinals), so cofinality supplies $\eta \in C$ with $\xi^{+} \le \eta$, whence $\xi \in \xi^{+} \subseteq \eta$ and $\xi \in \bigcup C$, giving $\lambda \subseteq \bigcup C$. This is the form in which the notion is used on this page, and it is exactly the hypothesis of the continuity clause of Monotonicity of ordinal $+$ and $\cdot$: strictly increasing and continuous in the right argument, weakly increasing in the left, with left cancellation, and the identities $0 + \beta = \beta$ and $1 \cdot \beta = \beta$.

• At $0$ and at successors the notion is degenerate. $\varnothing$ is cofinal in $0$, vacuously, and it is the only subset of $0$. If $\alpha = \delta^{+}$ then $\delta$ is the greatest element of $\alpha$ (Basic closure properties of ordinals, Trichotomy and well-ordering of the ordinals), so a subset is cofinal in $\alpha$ if and only if it contains $\delta$. The interesting case is the limit case, and that is where the notion is used.

• What is not defined at this point in the reading order. The cofinality $\operatorname{cf}(\alpha)$, the least order type of a cofinal subset, and the vocabulary of regular and singular cardinals, are not introduced here; they are introduced later, on Cardinal Arithmetic, Cofinality and the Alephs. Nothing on this page needs them: the boundedness theorem below is stated as "no at most countable subset is cofinal", which is a statement about subsets and not about a cardinal invariant.

• Cofinal is a property of the pair, not of the set. $\omega$ is cofinal in $\omega$ and bounded below $\omega + 1$. The ordinal must always be named.