Relationship between AC, WO and Zorns Lemma in ZF-PowersetAxiom of Choice and Order TypesAxiom of Choice in a weaker systemIs choice needed to establish the existence of idempotent ultrafilters?Does ZFC prove the universe is linearly orderable?Axiom of Choice and Number TheoryRelationship between fragments of the axiom of choice and the dependent choice principlesAbout the hypothesis of Zorn's lemmaZorn's lemma via Zermelo theoremRelation between the Axiom of Choice and a the existence of a hyperplane not containing a vectorHow is this fixed point theorem related to the axiom of choice?

Relationship between AC, WO and Zorns Lemma in ZF-Powerset


Axiom of Choice and Order TypesAxiom of Choice in a weaker systemIs choice needed to establish the existence of idempotent ultrafilters?Does ZFC prove the universe is linearly orderable?Axiom of Choice and Number TheoryRelationship between fragments of the axiom of choice and the dependent choice principlesAbout the hypothesis of Zorn's lemmaZorn's lemma via Zermelo theoremRelation between the Axiom of Choice and a the existence of a hyperplane not containing a vectorHow is this fixed point theorem related to the axiom of choice?













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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.










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    $begingroup$


    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.










    share|cite|improve this question









    New contributor




    Hannes Jakob is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
    Check out our Code of Conduct.







    $endgroup$














      5












      5








      5





      $begingroup$


      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.










      share|cite|improve this question









      New contributor




      Hannes Jakob is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.







      $endgroup$




      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.







      set-theory axiom-of-choice independence-results






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      Hannes Jakob is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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      share|cite|improve this question









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      edited 13 hours ago









      Martin Sleziak

      3,13032231




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      asked 13 hours ago









      Hannes JakobHannes Jakob

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          1 Answer
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          8












          $begingroup$

          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.







          share|cite|improve this answer









          $endgroup$








          • 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
            4 hours ago











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          1 Answer
          1






          active

          oldest

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          active

          oldest

          votes






          active

          oldest

          votes









          8












          $begingroup$

          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.







          share|cite|improve this answer









          $endgroup$








          • 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
            4 hours ago















          8












          $begingroup$

          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.







          share|cite|improve this answer









          $endgroup$








          • 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
            4 hours ago













          8












          8








          8





          $begingroup$

          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.







          share|cite|improve this answer









          $endgroup$



          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.








          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered 13 hours ago









          Asaf KaragilaAsaf Karagila

          22k681187




          22k681187







          • 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
            4 hours ago












          • 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
            4 hours ago







          1




          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
          4 hours ago




          $begingroup$
          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
          4 hours ago










          Hannes Jakob is a new contributor. Be nice, and check out our Code of Conduct.









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