What would be some major consequences of the inconsistency of ZFC?

Question

Update (21st April, 2019). Removed the reference / initial trigger behind my question (please see comment thread below for the reasons). Am retaining, of course, the actual question, noted both in the title as well as in the post below.

The key questions of this post are the following:

1. How "disastrous" would inconsistency of ZFC really be?

2. A slightly more refined question is: what would be the major consequences of different types of alleged inconsistencies in ZFC?

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Old material (the "no-longer relevant" part of the question).

I was happily surfing the arXiv, when I was jolted by the following paper:

Inconsistency of the Zermelo-Fraenkel set theory with the axiom of choice and its effects on the computational complexity by M. Kim, Mar. 2012.

Abstract. This paper exposes a contradiction in the Zermelo-Fraenkel set theory with the axiom of choice (ZFC). While Godel's incompleteness theorems state that a consistent system cannot prove its consistency, they do not eliminate proofs using a stronger system or methods that are outside the scope of the system. The paper shows that the cardinalities of infinite sets are uncontrollable and contradictory. The paper then states that Peano arithmetic, or first-order arithmetic, is inconsistent if all of the axioms and axiom schema assumed in the ZFC system are taken as being true, showing that ZFC is inconsistent. The paper then exposes some consequences that are in the scope of the computational complexity theory.

Now this seems to be a very major claim, and I lack the background to be able to judge if the claim is true, or there is some subtle or even obvious defect in the paper's arguments. But picking on this paper itself is not the purpose of my question.

If you feel that my questions might not admit "clearly right" answers, I will be happy to make this post CW.

Answer 1

I'm confident that ZFC is consistent, but one can imagine an inconsistency. Like François said, it would probably be handled pretty well. I'd divide the possibilities into four cases:

1. A technicality, like separation vs. comprehension in ZFC. This would be an important thing to get right, but it would have little impact on the theorems mathematicians prove. (For example, Frege's system was inconsistent, but his mistake didn't propagate.)

2. A topic requiring serious clarification, like infinitesimals in the 1600's. The intution was right, but it took some genuine work to turn this intuition into actual theorems with rigorous proofs.

3. A topic that fundamentally cannot be clarified, where some part of mathematics just turns out to be defective. For example, imagine if cardinals beyond $\aleph_0$ were inherently self-contradictory, and no clarification could save them. This would require huge modifications to set theory.

4. It could turn out that we have no idea what any of mathematics really means. For example, if Peano Arithmetic were inconsistent, then it would call into question the whole axiomatic approach to mathematics. It would be tantamount to saying that the natural numbers as we understand them do not exist. (Some parts of the axiomatic approach could still survive, but I don't think it would be wise to trust anything if we couldn't even get the consistency of PA right.)

My feeling is that 1 is very unlikely, 2 would be among the biggest shocks in the history of mathematics, 3 is difficult to imagine, and 4 is so extreme that if I read a proof of the inconsistency of PA, I'd be more likely to decide that I had gone crazy than that PA was actually inconsistent.

Answer 2

(This began as a comment to Henry Cohn’s answer, which which I mostly agree, but I’m over the character limit.) I think the specifics of the example in case 4 actually overlap with 1; an easily-remedied technical inconsistency in the Peano Axioms mightn’t call into question the whole axiomatic approach to mathematics, but perhaps only some details of it, such as using first-order rather than higher-order logic. To reverse your mention of Frege, in the nineteenth century, it wouldn’t have been that surprising if Frege’s higher-order arithmetic had been the only workable approach, and the first-order axiomatization failed. (Boolos made some good arguments in this context about the value of higher-order logic.) Indeed, the existence of non-standard models of the Peano Axioms proves that they don’t capture our intuition about the integers, so it might not be the end of formal mathematics if those particular axioms failed in the other direction as well.

    Let me note that I’m just addressing the hypothetical reaction to the inconsistency of PA, not the actual plausibility of it, and certainly not the extended consequences given, for instance, the available consistency proofs. See, for instance, Harvey Friedman’s rejoinder to Angus Macintyre’s Vienna lecture http://www.cs.nyu.edu/pipermail/fom/2011-June/015572.html. Angus was my undergraduate supervisor, but I don’t necessarily disagree with Friedman’s points. (Though I do disagree with his capitalization of Angus’s surname, a frequent error :-)

    My answer to the original question would be analogous. I don’t see any reason to believe in the inconsistency of ZFC, but the details of the axiomatization don’t exactly capture the underlying intuition: The iterative conception of set only justifies Replacement for formulæ which quantify over already-constructed levels of the cumulative hierarchy. Randall Holmes has argued that the theory justified by the iterative conception is actually Zermelo Set Theory with Σ2 replacement. (http: //math.boisestate.edu/~holmes/holmes/sigma1slides.ps. According to Professor Holmes, “this contain[s] an error, which Kanamori pointed out to me and which I know how to fix.”) So the reaction to an inconsistency in ZFC might be more like the adjustments for the Russell Paradox to Frege’s arithmetic than to his set theory—Frege’s development of arithmetic holds up remarkably well, with a few adjustments to his formalism: see, e.g., John Burgess’s Fixing Frege and Richard G. Heck’s Reading Frege's Grundgesetze.

Answer 3

Notice that you can drop the axiom of replacement or replace it by a weaker reflection principle. Without this axiom you have less consistency strength—it might still be consistent even if ZFC is not consistent. You would not loose that much. Of course many results in set theory would become meaningless (regarding various axioms, large cardinals etc.), but in most (not all, even when excluding set theory) situations in mathematics this axiom is not that important.

Answer 4

To misquote Scott Aaronson,[1] every paper ever published in Annals of Mathematics would have to be withdrawn ("the theorems are still true, but so are their negations").

[1] http://www.scottaaronson.com/papers/pnp.pdf p. 3