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!
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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 BlassCommented Jul 19, 2019 at 19:16
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5$\begingroup$ Every attempt to prove any mathematical fact is implicitly an attempt to prove inconsistency of ZF(C). $\endgroup$– YCorCommented Jul 20, 2019 at 8:41
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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 ♦Commented Jul 20, 2019 at 9:47
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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.
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2$\begingroup$ This might also be relevant mathoverflow.net/questions/36693/… $\endgroup$– QfwfqCommented Jul 19, 2019 at 19:20