# Is there a name for the smallest ordinal $\alpha$ such that $X \subseteq \alpha$

Let $X$ be a set of ordinals.

If $X$ has no largest element, then $\sup X \notin X \subseteq \sup X,$ and $\sup X$ is the smallest ordinal $\alpha$ such that $X \subseteq \alpha$.

On the other hand, if $X$ has a largest element $\max X$, then $\max X = \sup X \in X \nsubseteq \sup X,$ and the smallest ordinal $\alpha$ such that $X \subseteq \alpha$ is $\sup X + 1$.

Is there any way to express "the smallest ordinal $\alpha$ such that $X \subseteq \alpha$" that works in both cases?

One possibility would be $\sup \{ \beta + 1 : \beta \in X \},$ but I'm looking for something more concise or elegant than that.

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I've seen $\mathrm{ssup}(X)$, the strict supremum of $X$. I'm not a fan though. – François G. Dorais Dec 26 '12 at 6:40
I have seen $\sup^+(X)$ used in this context. – Asaf Karagila Dec 26 '12 at 7:43

$\mathrm{rank}(X)$