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!

All three are variations of the Axiom of Choice: https://en.wikipedia.org/wiki/Axiom_of_choice
There is no ultimate proof because it is an independent axiom.
