Question

Is it consistent without the axiom of choice that every permutation of some infinite set have fixed points?

score 253 · Answer 1 · 2015-05-06 20:40:56

253

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Is it consistent without the axiom of choice that every permutation of some infinite set have fixed points?

Published on 06 May 2015 - 20:40

score 251 · Answer 2 · 2015-05-07 18:31:38

251

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Can we prove AC from the statement "There is no $\aleph$ cardinal strictly between $\operatorname{CARD}(X)$ and $\operatorname{CARD}(2^X)$"?

Published on 07 May 2015 - 18:31

score 1.9k · Answer 3 · 2015-05-11 12:48:24

1.9k

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Does the proof of Bolzano-Weierstrass theorem require axiom of choice?

Published on 11 May 2015 - 12:48

score 188 · Answer 4 · 2026-04-01 03:41:25

188

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Axiom of choice : continuous function and uniformly continuous

Published on 01 Apr 2026 - 3:41

score 752 · Answer 5 · 2015-05-11 18:11:15

752

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Does a direct sum decomposition of an infinite-dimensional vector space require Zorn's lemma?

Published on 11 May 2015 - 18:11

score 70 · Answer 6 · 2026-03-30 04:56:33

70

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The existence of the sequence corresponding to some asymptotic sequence

Published on 30 Mar 2026 - 4:56

score 311 · Answer 7 · 2015-05-14 16:01:36

311

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Can we prove, without axiom of choice, that the set of all zero divisors (including $0$) of a commutative ring with unity contains a prime ideal?

Published on 14 May 2015 - 16:01

score 86 · Answer 8 · 2026-03-30 05:02:16

86

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Does "$(\exists f:A\twoheadrightarrow B)\implies(\exists f:B\hookrightarrow A)$" implies the axiom of choice?

Published on 30 Mar 2026 - 5:02

score 405 · Answer 9 · 2015-05-15 21:09:53

405

Views

How to prove that $\sf CH$ implies $2^{\aleph_0}=\aleph_1$

Published on 15 May 2015 - 21:09

score 502 · Answer 10 · 2026-03-30 07:28:20

502

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How to show countability of $\omega^\omega$ or $\epsilon_0$ in ZF?

Published on 30 Mar 2026 - 7:28

score 153 · Answer 11 · 2015-05-16 22:10:56

153

Views

If $|A|<|B|$ does $B$ surject onto $\aleph(A)$?

Published on 16 May 2015 - 22:10

score 28 · Answer 12 · 2015-05-17 18:27:48

28

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Is AC necessary to show that in metric spaces $x\in\operatorname{closure}(A)$ implies $\exists\{a_n\}_{n=1}^\infty\subseteq A$ s.t. $\lim a_n=x$?

Published on 17 May 2015 - 18:27

score 166 · Answer 13 · 2026-03-26 19:20:55

166

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Looking for explanation of Banach-Tarski Proof, preferably by visual methods "Video, Pictures, Diagrams..."

Published on 26 Mar 2026 - 19:20

score 583 · Answer 14 · 2015-05-19 15:38:41

583

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Choice function for collection of arbitrary finite sets. AC required?

Published on 19 May 2015 - 15:38

score 1.2k · Answer 15 · 2015-05-19 17:55:29

1.2k

Views

What's wrong with this proof of the inconsistency of the axiom of choice?

Published on 19 May 2015 - 17:55

List Question

Is it consistent without the axiom of choice that every permutation of some infinite set have fixed points?

Can we prove AC from the statement "There is no $\aleph$ cardinal strictly between $\operatorname{CARD}(X)$ and $\operatorname{CARD}(2^X)$"?

Does the proof of Bolzano-Weierstrass theorem require axiom of choice?

Axiom of choice : continuous function and uniformly continuous

Does a direct sum decomposition of an infinite-dimensional vector space require Zorn's lemma?

The existence of the sequence corresponding to some asymptotic sequence

Can we prove, without axiom of choice, that the set of all zero divisors (including $0$) of a commutative ring with unity contains a prime ideal?

Does "$(\exists f:A\twoheadrightarrow B)\implies(\exists f:B\hookrightarrow A)$" implies the axiom of choice?

How to prove that $\sf CH$ implies $2^{\aleph_0}=\aleph_1$

How to show countability of $\omega^\omega$ or $\epsilon_0$ in ZF?

If $|A|<|B|$ does $B$ surject onto $\aleph(A)$?

Is AC necessary to show that in metric spaces $x\in\operatorname{closure}(A)$ implies $\exists\{a_n\}_{n=1}^\infty\subseteq A$ s.t. $\lim a_n=x$?

Looking for explanation of Banach-Tarski Proof, preferably by visual methods "Video, Pictures, Diagrams..."

Choice function for collection of arbitrary finite sets. AC required?

What's wrong with this proof of the inconsistency of the axiom of choice?

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