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Let us say that someone was able to prove that $0=1$ using ZFC, thereby proving it inconsistent. What impact would this have on the study of meta-mathematics?

Most mathematicians would just move onto a different set theory, since most mathematics is not sensitive to the exact axioms being used.

Meta-mathematics, on the other hand, is. In particular, I'm talking about model theory, set theory, proof theory, etc... What results would become meaningless, and which could be salvaged. What other set theories could be used instead?

José Carlos Santos
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Christopher King
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  • https://math.stackexchange.com/questions/247026/what-would-happen-if-zfc-were-found-to-be-inconsistent – Asaf Karagila Jul 12 '17 at 16:12
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    What can be salvaged and what not depends on what sort of proof you get. If the proof is really a proof from PA, then a lot will break down. If the proof is using power set axiom, we still can get second-order arithmetic, or maybe even more. If the proof is due to Replacement axioms, then we can get even more. If the proof is due to the fact that first-order logic is itself inconsistent, then mathematics itself is in trouble. If the proof is just wrong, then we're not in a bad situation. – Asaf Karagila Jul 12 '17 at 16:14

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People will start paying more attention to results like Patey-Yokoyama on Ramsey phenomena, which show that a lot more mathematics than was thought earlier can done conservatively relative to a finitistic framework.

Another issue is the possible impact of a discovery of an inconsistency in ZFC on traditional beliefs in the existence of an intended model/intended interpretation of ZFC. Namely, in such a hypothetical situation of ZFC having turned out to be inconsistent, how will such beliefs evolve and what strategies will be developed to deflect the question "what is this supposed to have been an intended model of exactly".

Mikhail Katz
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    I don't really think this is accurate - ZFC is incredibly far from "finitistic" theories. Already, merely dropping replacement results in a vastly weaker system that is still ludicrously stronger than full second-order arithmetic. I don't think mathematicians would start paying particularly more attention to results at this level (although I personally find them very interesting). In particular, when you say "a lot more mathematics than was thought earlier can done conservatively relative to a finitistic framework" that's misleading: the vast majority of mathematics is well beyond that system. – Noah Schweber Jul 12 '17 at 17:35
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    E.g. if you want functions of natural numbers to have ranges, then you already jump above that. I don't think an inconsistency in ZFC would lead any mathematician to be skeptical of that principle. – Noah Schweber Jul 12 '17 at 17:36
  • @NoahSchweber, I saw somewhere that Simpson thinks about 70% of mathematics can be done within that system. – Mikhail Katz Jul 13 '17 at 07:17
  • I've never heard that claim, and in my opinion it's patently ridiculous: for instance, already the mere statement "the range of any function $\mathbb{N}\rightarrow\mathbb{N}$ is a set" isn't finitistic! – Noah Schweber Jul 23 '17 at 21:42