In regular ZF, AC, WO, and Zorn's Lemma are equivalent, but every proof I know (of the implication AC -> WO and AC -> Zorn) uses the axiom of choice on the powerset of X (where X is the Set which is to be well-ordered).

My question is, whether or not there is a proof of this equivalence that doesn't use the axiom of choice or whether there is a Model of ZF-Powerset in which AC holds but the well-ordering principle (or Zorn's Lemma) fails.


This is a classic theorem of Zarach, that it is consistent that ${\sf ZF}^-$ holds with the Axiom of Choice, but not every set can be well-ordered.

Zarach, Andrzej, Unions of ${\sf ZF}^-$models which are themselves ${\sf ZF}^-$ models, Logic colloquium ’80, Eur. Summer Meet., Prague 1980, Stud. Logic Found. Math. 108, 315-342 (1982). ZBL0524.03039.

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    $\begingroup$ In the quoted paper (p.338), Zarach credits Zbigniew Szczepaniak with first having demonstrated in 1979 that in the ZF^- context, the axiom of choice does not imply that every set can be well-ordered; and he acknowledges how Szcepaniak's work relates to his. $\endgroup$
    – Ali Enayat
    Apr 26 '19 at 19:24

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