Well Ordering Principle, Zorn's Lemma, Induction, etc. I know there are plenty of proofs online that show Induction implies well ordering, well ordering implies induction, zorn's lemma implies induction, etc. and what I seem to be getting is that in most courses professors introduce these by assuming one and proving the others from it, but is there an accessible proof of one that does not assume the others and is self-contained? I don't seem to be finding any. I hate the idea that you more or less have to assume one because the proof of any one of them without using the others is too unaccessible to undergraduate students. Thanks!

There are 3 well known equivalent statements:


*

*Axiom of choice

*Zorn's lemma

*Well ordering principle


Further, there is a tool of the set theory which does not require any of these, it is called transfinite recursion.
But I see these three "equivalent" statements a bit differently. Of course, they are equivalent, but you do not need anything advanced (I mean transfinite recursion, ordinals, etc.) to prove 1 from 2 or from 3. On the other hand, you need the transfinite recursion for proving 2 or 3 from 1. Moreover, 1 is really intuitive, there is a reason why to consider it as an axiom.
So in my view, 2 and 3 are just theorems of set theory which can be proved using the axiom of choice (as well as other axioms) and the transfinite recursion. And these two theorems happened to be so strong and general that they can prove each other and the original axiom.
