I want to find some necessary and sufficient conditions of a well-ordered set to be of order type $\omega$.
To be specific, let $A$ be a well-ordered set, if each element of $A$ except the smallest one has an immediate predecessor, then is the order type of $A$ "at most $\omega$"? (Selfishly by the words "at most $\omega$" I mean there is an order-preserving injection from $A$ to $\mathbb{Z}_+$.) I feel it is true intuitively, but I don't know how to find the order-preserving injection.
Could you tell me whether the proposition is true and how to prove it? Thanks a lot.
Yes. An uninteresting proof is trivial: If the order type is not at most $\omega$ then it is at least $\omega+1$, which means there is an element, namely $\omega$, which does not have an immediate predecessor.
Maybe slightly more interesting: Show by induction that every element of $A$ has only finitely many predecessors...