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I am stuck on the familiar question of how one thinks about models of ZFC without being circular. Concretely, if a set is defined to be anything that meets the axioms of ZFC, and a model of ZFC is a set, then how have we said anything of substance? Heads up, I've read through all answers to all of these already:

Models of set theory

https://mathoverflow.net/questions/23060/set-theory-and-model-theory/23077#23077

Removing sets from models of set theory

How can there be genuine models of set theory?

Crashcourse Models in Set Theory

Confusion about countable models of ZF set theory.

Unlike in some of the other posts, I am comfortable with model theorists interpreting the axioms of ZFC within ZFC in practice. However, I am wondering whether there is a common understanding about how one could give a meaning or semantics to ZFC externally. This might mean placing ZFC within a different formal framework, or it might mean needing to appeal to extra-mathematical entities. In the former case though we are probably just further begging the question. So my question is is there a standard way to assign a semantics to ZFC without interpreting it as sets which are themselves defined by ZFC?

Neil
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    I don't think this question deserves a downvote (and personally I've upvoted). These are subtle and interesting issues and I think the OP has presented their question reasonably clearly. – Noah Schweber Jul 29 '19 at 15:19
  • That said, here are a couple points where I'm not sure what you mean: (1) "I will typically consider a pair to be a function from the set {1,2} into some field" - (2) "we should probably still read this as saying "IF you have something to interpret the empty set to THEN you have..." rather than stating "There IS an empty set"." Moreover, your claim that the axioms of category theory don't include existence statements is false in at least some systems - e.g. consider SEAR. Can you give an example of a system in which your statement is true? – Noah Schweber Jul 29 '19 at 15:22
  • Thank you @NoahSchweber for the constructive criticism. I do need to learn the meta of this site still. I have edited the question with answers. – Neil Jul 29 '19 at 15:32
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    By the way I am writing up an attempted answer to this question - it may take me a bit, though. – Noah Schweber Jul 29 '19 at 15:34
  • @Noah did you check if this is a duplicate of previous questions? (Other than the ones linked, that is) – Asaf Karagila Jul 29 '19 at 15:51
  • @AsafKaragila I can't find any that I really think is a complete duplicate. Of course, some of this is my erring on the side of generosity, but I do actually think there's a bit more going on here. My search may have been flawed, though ... – Noah Schweber Jul 29 '19 at 15:51
  • https://math.stackexchange.com/questions/121128/when-does-the-set-enter-set-theory for example, @Noah – Asaf Karagila Jul 29 '19 at 15:52
  • At a glance I don't think that one's an exact duplicate, but give me a couple minutes to write why. Of course you can close on your own, but I'll throw out a minor vote to let me defend the question before doing so. – Noah Schweber Jul 29 '19 at 15:54
  • @Noah: I have no intention of doing that at the moment, rest assured. – Asaf Karagila Jul 29 '19 at 15:55
  • I think the new feature of this question is the sociological aspect. I think (although this isn't stated explicitly) that the OP is comfortable with the "formalist bulwark," but would reject that as an answer on the grounds that it doesn't actually match how we tend to think. A key clause here for me is: "What background knowledge do we fall back on, however, when describing sets? Naive set theory? Just our intuitive notion of a "collection"? Is there any agreement about this or is this kind of what the debate is all about?" (continued) – Noah Schweber Jul 29 '19 at 16:02
  • Another is "I know that if pressed on the matter, I could fall back onto Kuratowski pairs to recover pairs without circularity.... it's usually better to think of the pair (5, 6) as being a function from {1, 2} into, say R, where 1 gets mapped to 5 and 2 gets mapped to 6. The problem with this though, as stated, is that functions themselves are sets of pairs. So there is an apparent circularity." - the OP I think sees well that this is evidence that that view of ordered pairs is flawed, and that there's a way out (relative to our comfort with ZFC itself) but they don't view that as satisfying. – Noah Schweber Jul 29 '19 at 16:02
  • Actually, my real problem with this question is that it's too broad - it mixes questions about "mathematical reality" with sociological questions with an implicit question about what differences there are, re: foundational commitment, between set theory and category theory, and I don't yet see an easy "unifier" for them. So I could see closing this for that reason, and I think that the first aspect (and possibly the third aspect) above are duplicates. But the question of how this foundational question is handled in practice, and implicitly how disagreements on this are responded to, seems new. – Noah Schweber Jul 29 '19 at 16:04
  • @AsafKaragila Actually, the more I think about it the more I'm worried that I'm being overly generous here. I still don't think this is certainly a duplicate, but the more I read it the less clear it is to me whether I'm right that the OP's main question is sociological, and in particular I'm increasingly getting the sense that it's far too broad. I've paused writing my answer, and may close myself in the near future. That said I don't think this deserves downvotes, but I do think it may need serious overhauling by the OP. – Noah Schweber Jul 29 '19 at 16:21
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    To the OP: I think first of all you should drop the category theory aspect, it's going in a really different direction. Second, you need to clarify your main question: are you asking about what "extra-mathematical fallbacks" are used in practice to make sense of set models of set theories and how differences in/disagreements over them are handled, or are you asking how to interpret the apparent existence claims of the ZFC axioms without going formalist or Platonist, or are you asking something else? Third, the question should be much shorter - more commentary doesn't necessarily add clarity. – Noah Schweber Jul 29 '19 at 16:29
  • (And keep in mind that the previous versions of the question will be viewable through the edit history: if you delete some text, it's not actually lost, it's just not presented "up front.") – Noah Schweber Jul 29 '19 at 16:30
  • Okay that's fair. Thanks. – Neil Jul 29 '19 at 16:35
  • @NoahSchweber I've given it a shot. – Neil Jul 29 '19 at 17:18
  • @Neil I like this version much better, and I'll try to answer it when I have time (I'm too busy right now). – Noah Schweber Jul 29 '19 at 18:06

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