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In this answer, Arturo wrote:

I would think like Sebastian, that most "working mathematicians" didn't worry too much about Russell's paradox; much like they didn't worry too much about the fact that Calculus was not, originally, on solid logical foundation. Mathematics clearly worked, and the occasional antinomy or paradox was likely not a matter of interest or concern.

And I wondered: is there some mathematical concept today that is not in a solid logical foundation?

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Red Banana
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    This look very similar http://mathoverflow.net/questions/56677/what-notions-are-used-but-not-clearly-defined-in-modern-mathematics – Baby Dragon Apr 29 '13 at 16:53

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Generally there always exists a framework that gives solid logical ground (modulo inconsistency issues with the framework). But often this framework is a little more restrictive and a requires mathematicians to be a little more careful then they, in practice, are. So, while one can claim that such foundations exist the claim that mathematicians actually use these foundations is pretty sketchy.

For example most mathematicians would say that ZFC puts set theory, and hence pretty much the rest of math, on solid logical foundations. But there are issues with using ZFC to do category theory. Most mathematicians do not care if their categories are small and make no attempt to address this issue. Also the definition of a derived functor, for example, is usually presented in a way that is not quite correct if you use ZFC as your foundations.

Both of these issues can be fixed by taking ZFCU as your foundations, but this just pushes the laziness issues a bit farther down the road. For example, most mathematicians don't bother to check that what they are doing is independent of a choice of Grothendieck universe. They assume it is, and are generally right. But there do exist examples where this isn't the case.

Jim
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  • Jim: Could you give an example of where the particular choice of Grothendieck universe matters? – Hanno Jul 02 '13 at 07:40
  • The only reference I know of is Waterhouse, but it's not an easy example: http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.pjm/1102906018 – Jim Jul 03 '13 at 04:28
  • Thank you, that looks interesting! – Hanno Jul 05 '13 at 07:02
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Well, we don't know whether $\mathsf{ZFC}$ is consistent, so I guess modern set theory is on pretty shaky ground!

user642796
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    But can one have well-founded doubts about a well-founded universe? – Brian M. Scott Apr 29 '13 at 16:03
  • Then so is numer theory, since to prove that PA is consistent, we need set theory. – fgp Apr 29 '13 at 16:04
  • @Brian: That sounds like some sort of [gasp!] Gödelian Platonism! – user642796 Apr 29 '13 at 16:17
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    As long as it isn’t Aristotelean Plutonianism! (Sorry: old Usenet reference.) – Brian M. Scott Apr 29 '13 at 16:19
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    "Pretty shaky" seems to be an overstatement..at least AFAWK. – DonAntonio Apr 29 '13 at 16:21
  • @Brian: I think you mean Archimedean Plutonianism. – user642796 Apr 29 '13 at 16:22
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    @DonAntonio aside (for this admittedly acronym-naive user): what is AFAWK? "as far as we know?" "and for all we know?" Just guessing... – amWhy Apr 29 '13 at 16:23
  • Damn! I certainly did, and I thought that that’s what I was writing. – Brian M. Scott Apr 29 '13 at 16:30
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    http://mathoverflow.net/questions/40920/what-if-current-foundations-of-mathematics-are-inconsistent-closed – Baby Dragon Apr 29 '13 at 16:59
  • @DonAntonio: Overstatement? Yes, and that was partly the point. (Of course, when your working assumption is "Nothing will go wrong" and the only evidence you can ever really have in support of this assumption is the anecdotal "Nothing has gone wrong... yet... AFAWK... Shelah's prolificity notwithstanding..." we are clearly not on the same solid foundations as was hoped by Russell & Whitehead, or by Hilbert.) – user642796 Apr 29 '13 at 17:11
  • You rather imaginationless (?) individuals: AFAWK = As Far as Wahalulu Knocks...:) Now seriously: amWhy's on the money: I meant "AFAWK=as far as we know" . We should really forbid some acronyms. – DonAntonio Apr 29 '13 at 18:24
  • I know personally Saharon, one of the most awards-winning mathematicians of our time, and I don't think he'll get very impressed if ZFC will be proven inconsistent: either he'll jump down from some rather high bridge or else he'll continue on as if nothing had happened...I like that! – DonAntonio Apr 29 '13 at 18:27
  • I think this definition of "know" is too strong, since it excludes too many things that are reasonably considered "knowledge". See Why is it considered unlikely that there could be a contradiction in ZF/ZFC?. – user76284 Aug 20 '20 at 21:07