Is the supremum of an ordinal the next ordinal? I apologize for this naive question.
Let $\eta$ be an ordinal. Isn't the supremum of $\eta$ just $\eta+1$? If this is true, the supremum is only necessary if you conisder sets of ordinals.

There are two notions of supremum in sets ordinals, let $A$ be a set of ordinals:


*

*$\sup^+(A)=\sup\{\alpha+1\mid\alpha\in A\}$,

*$\sup(A) =\sup\{\alpha\mid\alpha\in A\}$.


Since most of the time we care about supremum below limit ordinals (eg. $A=\omega$) the notions coincide. If $A=\{\alpha\}$ then indeed $\sup(A)=\alpha$ and $\sup^+(A)=\alpha+1$. 
The reason there are two notions is that $\sup^+(A)$ is defined as $\min\{\alpha\in\mathrm{Ord}\mid A\subseteq\alpha\}$ and $\sup(A)=\bigcup A$.
Both of these notions are useful, and it is easy to see that if $A$ has no maximal element then these indeed coincide. However the distinction can be useful in successor ordinals from time to time.
