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From what I've understood, Naive Set Theory does not mix very well with classical logic because it eventually boils down every statement to contradiction.

This is not to say Naive Set Theory is strictly useless, as long as you're willing to work with more exotic logic.

Pretty much all of the mathematics I've learnt is based on, as far as I know, the ZFC axioms. I understand that there are other alternatives to ZFC, but perhaps, have we figured out a way to ensure that something like Russell's Paradox will not arise in an arbitrary axiomatic foundation for sets? (when "scrutinised" under classical logic)

Threnody
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    Basically, we are confident that there is none. Russell's Paradox was one of many paradoxes of original set theory, all discovered pretty soon. Since the axiomatization of set theory, more than a century ago, no new problem arised... – Mauro ALLEGRANZA May 03 '20 at 15:10
  • See also this post as well as this one. – Mauro ALLEGRANZA May 03 '20 at 15:24
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    Another reason for confidence that ZFC won't lead to a contradiction is that we have a "clear" notion of set, namely the cumulative hierarchy, for which the ZFC axioms are "clearly" true. If both of those occurrences of "clear" are really clear, then so is the consistency of ZFC. Unfortunately, although I consider those things clear, that's not a mathematical fact. To prove it, we'd need to use axioms that somehow go beyond ZFC. – Andreas Blass May 03 '20 at 17:28
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    @AndreasBlass, ZFC axioms are clearly true of cumulative hierarchies of an inaccessible height. We can indeed build up hierarchies that are not of inaccessible height in which ZFC axioms would be falsified. I know that usually when people speak of the cumulative hierarchy they mean the Von Neumann's one which is of inaccessible height. But I think this point must be emphasized, in order avoid confusion between ZFC axioms and various kinds of hierarchies that can also be appropriately labeled as cumulative (i.e., built in power stages). – Zuhair May 05 '20 at 09:24

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This is very close to a duplicate of various questions, but I can't find an exact dupe at the moment.

Ultimately we are not certain of this. Indeed, although it's very rare there have been quite excellent mathematicians who suspected that $\mathsf{ZFC}$ is inconsistent after all (such as the logician Jack Silver). And this is all leaving aside the second incompleteness theorem. Personally, while I am as certain of the consistency of first-order Peano arithmetic as I am of the fact that I have two hands, I would merely be profoundly disturbed to learn that $\mathsf{ZFC}$ is inconsistent.

Fortunately, $\mathsf{ZFC}$ is so ridiculously overpowered that its inconsistency wouldn't really spill over to the rest of mathematics too much. For the vast majority of mathematical practice, galactically weaker theories like $\mathsf{ZC}$ are enough.

Incidentally, the question of exactly how much "axiomatic overhead" is needed for various parts of mathematics is studied rigorously in Reverse Mathematics. It turns out that a huge amount of mathematics can be developed in $\Pi^1_1$-$\mathsf{CA_0}$, which is a tiny fragment of $\mathsf{Z_2}$ which is itself a tiny fragment of $\mathsf{ZC}$. The primary difficulty with RM's approach is the restricted language, which makes lots of "higher-type" mathematics (e.g. measure theory, topology, ...) hard or impossible to treat faithfully, but there has been some recent work in improving this situation (see e.g. here).

Noah Schweber
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