A small problem, but I want it to be clear.
I'm reading Jech's Set Theory and find the following statement:
Lemma 2.24. For all ordinals $\alpha$ and $\beta$, $\alpha+\beta$ and $\alpha\cdot\beta$ are, respectively, isomorphic to the sum and to the product of $\alpha$ and $\beta$.
We're interested in the sum here. Before this lemma, the definition of the sum of linear orders is given:
Definition 2.22. Let $(A,<_A)$ and $(B,<_B)$ be disjoint linearly ordered sets. The sum of these linear orders is the set $A\cup B$ with the ordering defined as follows: $x<y$ if and only if
(i) $x,y\in A$ and $x<_A y$, or
(ii) $x,y\in B$ and $x<_B y$, or
(iii) $x\in A$ and $y\in B$.
Notice that in this definition the disjointness of the sets is required. So is the definition on Wikipedia as well.
But aren't any two ordinals joint, since for two arbitrary ordinals $\alpha$ and $\beta$, either $\alpha\subseteq\beta$ or $\beta\subseteq\alpha$? So how can we talk about the sum of two ordinals? Does the lemma use a version of the definition without the disjointness condition?
Yes. That is why we talk about order type which is the unique ordinal which is isomorphic to the order defined on $A\cup B$. The ordinal $\alpha+\beta$ is exactly the ordinal isomorphic to the disjoint sum of $\alpha$ and $\beta$.