I'm following the proof offered by the book "Introduction to Set Theory" by K. Hrbacek of the claim "every well orderable set X is equipotent to a unique initial ordinal number" (Theorem 1.3 p. 130).
They start saying that, since a well ordered set is isomorphic to an ordinal number, X is equipotent to some ordinal number, then they talk about the least ordinal number equipotent to $X$; I don't understand how they deduce the existence of such ordinal. Since they emphasized the existence of an ordinal equipotent to X, I guess they are taking the least member of "the set of ordinals equipotent to X"; I know that every set whose elements are ordinals is well ordered (by the membership relation), then I'll agree of doing so but just after we prove the existence of such a set, and I still have no idea of how doing so.
Thank you in advance for your clarifications !
Let $\alpha$ be an ordinal equipotent to $X$. Then we can form the set
$$\{\xi\in\alpha+1:\xi\text{ is an ordinal equipotent to }X\}\,,$$
and since it’s a non-empty set of ordinals, it has a least element. Any ordinal equipotent to $X$ that is not in this set is larger than $\alpha$ and therefore certainly not the least ordinal equipotent to $X$.