[hopfenspergerj]

We might as well use the axiom that “you can make arbitrary choices at each stage of a transfinite induction”. It’s mostly pedagogical tradition to make students translate that into one of the classical forms of Choice in their proofs.

[dullcrisp]

Does transfinite induction without the ability to make arbitrary choices make sense? Or is the axiom just that you can do transfinite induction?

[seanhunter]

Zorn's lemma is specifically that there is a maximal element of an ordered set that is structured in a specific way. That seems a lot weaker to me than just being able to make a completely arbitrary choice. Am I wrong about that?

[toodlemcnoodle]

You are correct that it SEEMS that way. They are, in fact, equivalent under traditional mathematical hypotheses about how sets work.