Suppose $\alpha, \beta$ are ordinals, as ordinals are sets of ordinals, it seems that it is possible to construct an ordinal that contains $\alpha$ and $\beta$ as elements.
I am not able to construct such an ordinal so far. If it is possible, could someone give an explicit construction with justification? Thanks so much!
definition of ordinal:
transitive: $x ⊆ S$ for every $x ∈ S$.
A total order $≤$ on a set S is said to be a well-ordering if for every non-empty subset $A ⊆ S$, there exists an element $m ∈ A$ such that $ ∀x ∈ A,m\le x$
An ordinal number is a set that is transitive and is well-ordered by the relation $α ≤ β ⇔ α ∈ β \lor α = β. $