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?
Multi tool use
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?
$begingroup$
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.
set-theory axiom-of-choice independence-results
edited Apr 26 at 9:56
Martin Sleziak
3,13032231
asked Apr 26 at 9:54
Hannes JakobHannes Jakob
334
$endgroup$
add a comment |
$begingroup$
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.
set-theory axiom-of-choice independence-results
edited Apr 26 at 9:56
Martin Sleziak
3,13032231
asked Apr 26 at 9:54
Hannes JakobHannes Jakob
334
$endgroup$
add a comment |
$begingroup$
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.
set-theory axiom-of-choice independence-results
edited Apr 26 at 9:56
Martin Sleziak
3,13032231
asked Apr 26 at 9:54
Hannes JakobHannes Jakob
334
$endgroup$
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
set-theory axiom-of-choice independence-results
edited Apr 26 at 9:56
Martin Sleziak
3,13032231
asked Apr 26 at 9:54
Hannes JakobHannes Jakob
334
edited Apr 26 at 9:56
Martin Sleziak
3,13032231
asked Apr 26 at 9:54
Hannes JakobHannes Jakob
334
edited Apr 26 at 9:56
Martin Sleziak
3,13032231
edited Apr 26 at 9:56
Martin Sleziak
3,13032231
edited Apr 26 at 9:56
Martin Sleziak
3,13032231
3,13032231
asked Apr 26 at 9:54
Hannes JakobHannes Jakob
334
asked Apr 26 at 9:54
Hannes JakobHannes Jakob
334
asked Apr 26 at 9:54
Hannes JakobHannes Jakob
334
334
add a comment |
add a comment |
1 Answer
1
active
oldest
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$begingroup$
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.
answered Apr 26 at 10:10
Asaf KaragilaAsaf Karagila
22k681187
$endgroup$
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
add a comment |
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1 Answer
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1 Answer
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active
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$begingroup$
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.
answered Apr 26 at 10:10
Asaf KaragilaAsaf Karagila
22k681187
$endgroup$
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
add a comment |
$begingroup$
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.
answered Apr 26 at 10:10
Asaf KaragilaAsaf Karagila
22k681187
$endgroup$
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
add a comment |
$begingroup$
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.
answered Apr 26 at 10:10
Asaf KaragilaAsaf Karagila
22k681187
$endgroup$
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.
answered Apr 26 at 10:10
Asaf KaragilaAsaf Karagila
22k681187
answered Apr 26 at 10:10
Asaf KaragilaAsaf Karagila
22k681187
answered Apr 26 at 10:10
Asaf KaragilaAsaf Karagila
22k681187
answered Apr 26 at 10:10
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
Apr 26 at 19:24
add a comment |
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
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
Apr 26 at 19:24
$begingroup$
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
add a comment |
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