6
$\begingroup$

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 Zorns Lemma) fails.

$\endgroup$
10
$\begingroup$

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.

$\endgroup$
  • 1
    $\begingroup$ 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 at 19:24

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.