I'm struggling to understand a couple of steps in this proof showing that every well-ordered set is order isomorphic to an ordinal. Calling the following steps
first and second respectively, my issues are:
- First step: it seems to me that $S_x = (S_a)_x$ follows from $$S_x = \{ z\in S : z\prec x\} = \{ z\in S : z\prec x \land z\prec a\} = \{z\in S_a : z\prec x\} = (S_a)_x$$ as opposed from the definition of an ordinal.
- Second step: we have only showed that every proper segment $S_x$ of $S_a$ is order isomorphic to an ordinal i.e. there is still one segment of $S_a$ we have not shown to be order isomorphic to an ordinal, namely $S_a$ itself.
For the first step, you're absolutely right... that step has nothing to do with the definition of an ordinal and is a general statement about initial segments of ordered sets.
For the second step, note the lemma they're appealing to (whose proof is really where all the action is) says that a well-ordered set $S$ is isomorphic to an ordinal iff for all $x\in S,$ $S_x$ is isomorphic to an ordinal. So since $a\notin S_a,$ you don't need show anything for $S_a.$