Supplementary Exercises*: Well-Ordering

1. The general principle of recursive definition.
2. The maximum principle is equivalent to the well-ordering theorem.
3. The choice axiom is equivalent to the well-ordering theorem.

...one can construct an uncountable well-ordered set, and hence the minimal uncountable well-ordered set, by an explicit construction that does not use the choice axiom. However, this result is less interesting than it might appear. The crucial property of $S_{\Omega}$ , the one we use repeatedly, is the fact that every countable subset of $S_{\Omega}$ has an upper bound in $S_{\Omega}$ . That fact depends, in turn, on the fact that a countable union of countable sets is countable. And the proof of that result... involves an infinite number of arbitrary choices — that is, it depends on the choice axiom.
Said differently, without the choice axiom we may be able to construct the minimal uncountable well-ordered set, but we can’t use it for anything!