Can Well Ordering Theorem Be Proved Without the Axiom of Power Set?

Question

Can it be proved in ZFC - Pow (ZFC excluding the Axiom of Power Set) that Well Ordering Theorem holds? I have seen several proofs of Well Ordering Theorem in ZFC (including Zermelo's original one in 1904, Zermelo's in 1908), all involving choosing a choice function defined on the power set of the given set which we are going to well-order (To be more precise, the proof given by Wikipedia does not involve such a choice function, but it's deduced from Zorn's Lemma, and anyway it still requires the use of power sets.). Therefore, these proofs rely on the Axiom of Power Set, which makes me wonder whether such a proof is still possible without Power Set. Can anybody give such a proof (if possible) or otherwise construct a model of ZFC - Pow in which the Well Ordering Theorem fails?

Answer 1

No, it cannot be proved that every set can be well-ordered.

You can find details, and much much more, in the following paper.

Gitman, Victoria; Hamkins, Joel David; Johnstone, Thomas A., What is the theory $\mathsf{ZFC}$ without power set?, ZBL06642525, arXiv.