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As far as I know, no inconsistency is known to descend from the Zermelo-Fraenkel (ZFC) axioms. Question: historically, has a surprising inconsistency been found to descend from any comparably simple, seriously proposed, broadly credited system of axioms?

Another way to ask the question: when it comes to set theory, has the collective intuition of skilled mathematicians ever failed them?

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    $\begingroup$ There seems to be quite a wide gap between your two versions of the question. It only takes one person to seriously propose a system of mathematics, not mathematicians collectively. $\endgroup$ – Eric Wofsey Mar 16 at 19:03
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    $\begingroup$ Early ("naïve") set theory often took as an axiom that the collection of set-theoretic objects satisfying a formula would be a set. Russell's Paradox showed that this was not so! Indeed, a contradiction arises in that case from just the Axiom Schema of (unrestricted) Comprehension! $\endgroup$ – Cameron Buie Mar 16 at 19:03
  • $\begingroup$ @EricWofsey In my first version, I should have said, "broadly credited." I will edit. $\endgroup$ – thb Mar 16 at 19:09
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    $\begingroup$ I don't know of any (besides Frege's set theory); the closest thing I can think of is Reinhardt cardinals. I believe there were some people (Reinhardt himself?) who believed that ZFC + "There is a Reinhardt cardinal" would be consistent. However, my impression that the vast majority of the set-theoretic community was highly skeptical, and indeed $(i)$ it wasn't long before an inconsistency was found and $(ii)$ while substantially harder to establish than Russell's paradox, it still wasn't that complicated. $\endgroup$ – Noah Schweber Mar 16 at 19:16
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    $\begingroup$ (If you're interested, the original paper is quite readable; the key point is Theorem 1.12, which only relies on Lemma 1.11; and in fact we only need the cofinality-omega instance, which has a much simpler proof. The whole argument is then less than a page!) Embarrassingly, when I read the relevant part of Reinhardt's thesis I was totally convinced! So I don't know about mathematicians' intuition, but mine ... I believe the community suspects that (1) ZF+Reinhardt is also inconsistent but (2) the proof will be hard, and the first "deep inconsistency." $\endgroup$ – Noah Schweber Mar 16 at 19:23

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