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.

strict supremumof $X$. I'm not a fan though. – François G. Dorais♦ Dec 26 '12 at 6:40