Which failed attempts have there been to find a contradiction in ZFC or ZF?Nelson's program to show inconsistency of ZFSilver's approach to the inconsistency of $ZFC$Is there a computable model of ZFC?Interpretation of the Second Incompleteness TheoremMartin's Axiom and Determinacy-axioms: independence resultsIf ZFC has a transitive model, does it have one of arbitrary size?A question about sentences in the language of first order ZFC which assert the existence of cardinal numbersWholeness Axiom and Ultimate LIs there a proof (maybe formulated by Feferman) which says that a proof about the (in)consistency of ZFC is unachievable?Silver's approach to the inconsistency of $ZFC$Experiments physically performable in a finite amount of time whose results are independent of ZFCExistence of a model of ZFC in which the natural numbers are really the natural numbers
Which failed attempts have there been to find a contradiction in ZFC or ZF?
Nelson's program to show inconsistency of ZFSilver's approach to the inconsistency of $ZFC$Is there a computable model of ZFC?Interpretation of the Second Incompleteness TheoremMartin's Axiom and Determinacy-axioms: independence resultsIf ZFC has a transitive model, does it have one of arbitrary size?A question about sentences in the language of first order ZFC which assert the existence of cardinal numbersWholeness Axiom and Ultimate LIs there a proof (maybe formulated by Feferman) which says that a proof about the (in)consistency of ZFC is unachievable?Silver's approach to the inconsistency of $ZFC$Experiments physically performable in a finite amount of time whose results are independent of ZFCExistence of a model of ZFC in which the natural numbers are really the natural numbers
$begingroup$
Can you provide any failed attempts to prove that ZF or ZFC to be inconsistent?
References to articles in the literature if there are any will be much appreciated.
Thanks!
set-theory
asked Jul 19 at 18:53
AlanAlan
6554 silver badges15 bronze badges
$endgroup$
add a comment |
$begingroup$
Can you provide any failed attempts to prove that ZF or ZFC to be inconsistent?
References to articles in the literature if there are any will be much appreciated.
Thanks!
set-theory
asked Jul 19 at 18:53
AlanAlan
6554 silver badges15 bronze badges
$endgroup$
12
$begingroup$
I assume you want attempts by competent mathematicians (not cranks). Competent set theorists who doubt the consistency of ZFC are rare, and ones who have seriously sought a contradiction are even rarer. The only example that comes to mind is Jack Silver, who didn't publish failed attempts, but did publish a lot of good set theory arising from those attempts. See mathoverflow.net/questions/260944 for a little information about that.
$endgroup$
– Andreas Blass
Jul 19 at 19:16
3
$begingroup$
Every attempt to prove any mathematical fact is implicitly an attempt to prove inconsistency of ZF(C).
$endgroup$
– YCor
Jul 20 at 8:41
1
$begingroup$
One could, at a stretch, classify Gödel's consistency proof for AC and CH with ZF as double checking there isn't an inconsistency arising from those two axioms relative to the rest of ZF.
$endgroup$
– David Roberts
Jul 20 at 9:47
add a comment |
$begingroup$
Can you provide any failed attempts to prove that ZF or ZFC to be inconsistent?
References to articles in the literature if there are any will be much appreciated.
Thanks!
set-theory
asked Jul 19 at 18:53
AlanAlan
6554 silver badges15 bronze badges
$endgroup$
Can you provide any failed attempts to prove that ZF or ZFC to be inconsistent?
References to articles in the literature if there are any will be much appreciated.
Thanks!
set-theory
set-theory
asked Jul 19 at 18:53
AlanAlan
6554 silver badges15 bronze badges
asked Jul 19 at 18:53
AlanAlan
6554 silver badges15 bronze badges
asked Jul 19 at 18:53
AlanAlan
6554 silver badges15 bronze badges
asked Jul 19 at 18:53
AlanAlan
6554 silver badges15 bronze badges
asked Jul 19 at 18:53
AlanAlan
6554 silver badges15 bronze badges
6554 silver badges15 bronze badges
12
$begingroup$
I assume you want attempts by competent mathematicians (not cranks). Competent set theorists who doubt the consistency of ZFC are rare, and ones who have seriously sought a contradiction are even rarer. The only example that comes to mind is Jack Silver, who didn't publish failed attempts, but did publish a lot of good set theory arising from those attempts. See mathoverflow.net/questions/260944 for a little information about that.
$endgroup$
– Andreas Blass
Jul 19 at 19:16
3
$begingroup$
Every attempt to prove any mathematical fact is implicitly an attempt to prove inconsistency of ZF(C).
$endgroup$
– YCor
Jul 20 at 8:41
1
$begingroup$
One could, at a stretch, classify Gödel's consistency proof for AC and CH with ZF as double checking there isn't an inconsistency arising from those two axioms relative to the rest of ZF.
$endgroup$
– David Roberts
Jul 20 at 9:47
add a comment |
12
$begingroup$
I assume you want attempts by competent mathematicians (not cranks). Competent set theorists who doubt the consistency of ZFC are rare, and ones who have seriously sought a contradiction are even rarer. The only example that comes to mind is Jack Silver, who didn't publish failed attempts, but did publish a lot of good set theory arising from those attempts. See mathoverflow.net/questions/260944 for a little information about that.
$endgroup$
– Andreas Blass
Jul 19 at 19:16
3
$begingroup$
Every attempt to prove any mathematical fact is implicitly an attempt to prove inconsistency of ZF(C).
$endgroup$
– YCor
Jul 20 at 8:41
1
$begingroup$
One could, at a stretch, classify Gödel's consistency proof for AC and CH with ZF as double checking there isn't an inconsistency arising from those two axioms relative to the rest of ZF.
$endgroup$
– David Roberts
Jul 20 at 9:47
12
12
$begingroup$
I assume you want attempts by competent mathematicians (not cranks). Competent set theorists who doubt the consistency of ZFC are rare, and ones who have seriously sought a contradiction are even rarer. The only example that comes to mind is Jack Silver, who didn't publish failed attempts, but did publish a lot of good set theory arising from those attempts. See mathoverflow.net/questions/260944 for a little information about that.
$endgroup$
– Andreas Blass
Jul 19 at 19:16
$begingroup$
I assume you want attempts by competent mathematicians (not cranks). Competent set theorists who doubt the consistency of ZFC are rare, and ones who have seriously sought a contradiction are even rarer. The only example that comes to mind is Jack Silver, who didn't publish failed attempts, but did publish a lot of good set theory arising from those attempts. See mathoverflow.net/questions/260944 for a little information about that.
$endgroup$
– Andreas Blass
Jul 19 at 19:16
3
3
$begingroup$
Every attempt to prove any mathematical fact is implicitly an attempt to prove inconsistency of ZF(C).
$endgroup$
– YCor
Jul 20 at 8:41
$begingroup$
Every attempt to prove any mathematical fact is implicitly an attempt to prove inconsistency of ZF(C).
$endgroup$
– YCor
Jul 20 at 8:41
1
1
$begingroup$
One could, at a stretch, classify Gödel's consistency proof for AC and CH with ZF as double checking there isn't an inconsistency arising from those two axioms relative to the rest of ZF.
$endgroup$
– David Roberts
Jul 20 at 9:47
$begingroup$
One could, at a stretch, classify Gödel's consistency proof for AC and CH with ZF as double checking there isn't an inconsistency arising from those two axioms relative to the rest of ZF.
$endgroup$
– David Roberts
Jul 20 at 9:47
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
I think the most noticeable one was Nelson's attempt to prove the inconsistency of primitive recursive arithmetic. Terence Tao found a mistake in the proof, but Nelson's attempt was posthumously uploaded to arxiv (https://arxiv.org/pdf/1509.09209.pdf), together with an introduction by Sarah Jones Nelson and an afterword by Sam Buss and Terence Tao himself.
Nelson was amongst the very few serious mathematicians who supported the view that arithmetic was indeed inconsistent, based on his ultrafinitist philosophy.
edited Jul 20 at 8:33
Nathaniel
5542 silver badges18 bronze badges
answered Jul 19 at 19:19
godeliangodelian
2,7241 gold badge23 silver badges25 bronze badges
$endgroup$
2
$begingroup$
This might also be relevant mathoverflow.net/questions/36693/…
$endgroup$
– Qfwfq
Jul 19 at 19:20
add a comment |
Your Answer
StackExchange.ready(function()
var channelOptions =
tags: "".split(" "),
id: "504"
;
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function()
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled)
StackExchange.using("snippets", function()
createEditor();
);
else
createEditor();
);
function createEditor()
StackExchange.prepareEditor(
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader:
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
,
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
);
);
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f336545%2fwhich-failed-attempts-have-there-been-to-find-a-contradiction-in-zfc-or-zf%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
I think the most noticeable one was Nelson's attempt to prove the inconsistency of primitive recursive arithmetic. Terence Tao found a mistake in the proof, but Nelson's attempt was posthumously uploaded to arxiv (https://arxiv.org/pdf/1509.09209.pdf), together with an introduction by Sarah Jones Nelson and an afterword by Sam Buss and Terence Tao himself.
Nelson was amongst the very few serious mathematicians who supported the view that arithmetic was indeed inconsistent, based on his ultrafinitist philosophy.
edited Jul 20 at 8:33
Nathaniel
5542 silver badges18 bronze badges
answered Jul 19 at 19:19
godeliangodelian
2,7241 gold badge23 silver badges25 bronze badges
$endgroup$
2
$begingroup$
This might also be relevant mathoverflow.net/questions/36693/…
$endgroup$
– Qfwfq
Jul 19 at 19:20
add a comment |
$begingroup$
I think the most noticeable one was Nelson's attempt to prove the inconsistency of primitive recursive arithmetic. Terence Tao found a mistake in the proof, but Nelson's attempt was posthumously uploaded to arxiv (https://arxiv.org/pdf/1509.09209.pdf), together with an introduction by Sarah Jones Nelson and an afterword by Sam Buss and Terence Tao himself.
Nelson was amongst the very few serious mathematicians who supported the view that arithmetic was indeed inconsistent, based on his ultrafinitist philosophy.
edited Jul 20 at 8:33
Nathaniel
5542 silver badges18 bronze badges
answered Jul 19 at 19:19
godeliangodelian
2,7241 gold badge23 silver badges25 bronze badges
$endgroup$
2
$begingroup$
This might also be relevant mathoverflow.net/questions/36693/…
$endgroup$
– Qfwfq
Jul 19 at 19:20
add a comment |
$begingroup$
I think the most noticeable one was Nelson's attempt to prove the inconsistency of primitive recursive arithmetic. Terence Tao found a mistake in the proof, but Nelson's attempt was posthumously uploaded to arxiv (https://arxiv.org/pdf/1509.09209.pdf), together with an introduction by Sarah Jones Nelson and an afterword by Sam Buss and Terence Tao himself.
Nelson was amongst the very few serious mathematicians who supported the view that arithmetic was indeed inconsistent, based on his ultrafinitist philosophy.
edited Jul 20 at 8:33
Nathaniel
5542 silver badges18 bronze badges
answered Jul 19 at 19:19
godeliangodelian
2,7241 gold badge23 silver badges25 bronze badges
$endgroup$
I think the most noticeable one was Nelson's attempt to prove the inconsistency of primitive recursive arithmetic. Terence Tao found a mistake in the proof, but Nelson's attempt was posthumously uploaded to arxiv (https://arxiv.org/pdf/1509.09209.pdf), together with an introduction by Sarah Jones Nelson and an afterword by Sam Buss and Terence Tao himself.
Nelson was amongst the very few serious mathematicians who supported the view that arithmetic was indeed inconsistent, based on his ultrafinitist philosophy.
edited Jul 20 at 8:33
Nathaniel
5542 silver badges18 bronze badges
answered Jul 19 at 19:19
godeliangodelian
2,7241 gold badge23 silver badges25 bronze badges
edited Jul 20 at 8:33
Nathaniel
5542 silver badges18 bronze badges
edited Jul 20 at 8:33
Nathaniel
5542 silver badges18 bronze badges
edited Jul 20 at 8:33
Nathaniel
5542 silver badges18 bronze badges
5542 silver badges18 bronze badges
answered Jul 19 at 19:19
godeliangodelian
2,7241 gold badge23 silver badges25 bronze badges
answered Jul 19 at 19:19
godeliangodelian
2,7241 gold badge23 silver badges25 bronze badges
answered Jul 19 at 19:19
godeliangodelian
2,7241 gold badge23 silver badges25 bronze badges
2,7241 gold badge23 silver badges25 bronze badges
2
$begingroup$
This might also be relevant mathoverflow.net/questions/36693/…
$endgroup$
– Qfwfq
Jul 19 at 19:20
add a comment |
2
$begingroup$
This might also be relevant mathoverflow.net/questions/36693/…
$endgroup$
– Qfwfq
Jul 19 at 19:20
2
2
$begingroup$
This might also be relevant mathoverflow.net/questions/36693/…
$endgroup$
– Qfwfq
Jul 19 at 19:20
$begingroup$
This might also be relevant mathoverflow.net/questions/36693/…
$endgroup$
– Qfwfq
Jul 19 at 19:20
add a comment |
Thanks for contributing an answer to MathOverflow!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f336545%2fwhich-failed-attempts-have-there-been-to-find-a-contradiction-in-zfc-or-zf%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
12
$begingroup$
I assume you want attempts by competent mathematicians (not cranks). Competent set theorists who doubt the consistency of ZFC are rare, and ones who have seriously sought a contradiction are even rarer. The only example that comes to mind is Jack Silver, who didn't publish failed attempts, but did publish a lot of good set theory arising from those attempts. See mathoverflow.net/questions/260944 for a little information about that.
$endgroup$
– Andreas Blass
Jul 19 at 19:16
3
$begingroup$
Every attempt to prove any mathematical fact is implicitly an attempt to prove inconsistency of ZF(C).
$endgroup$
– YCor
Jul 20 at 8:41
1
$begingroup$
One could, at a stretch, classify Gödel's consistency proof for AC and CH with ZF as double checking there isn't an inconsistency arising from those two axioms relative to the rest of ZF.
$endgroup$
– David Roberts
Jul 20 at 9:47