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This is more of a question about philosophy of mathematics. There is an apparent circularity in set theory and logic, in the means that we use set theory to define model theory, and we use results from model theory for proofs in set theory. And, analogously, we use set theory to prove facts from logic (unique readability, for example) and we define set theory using logic. I feel that we can reduce all this notions to a single syntactical idea of logical consequence, and from there we can define things in a non-circular way. However, I can't seem to make this notions all fit together in a logical, foundational manner. Does anyone know of an article or book that addresses this issue?

Greg Nisbet
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Nuntractatuses Amável
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  • I’m sure this is more or less covered in the duplicates, but saying you “use set theory” to prove simple stuff about syntax is like saying you “use algebraic geometry” to prove two lines intersect at at most one point. – spaceisdarkgreen Dec 20 '18 at 02:29
  • ok, maybe it was a bad example. but take lowenheim-skloem theorem, for example. It is about first order theories and is proven using set theory, which is itself a first order theory. – Nuntractatuses Amável Dec 20 '18 at 03:11
  • Yes, that does use set theory, more or less. But note it is also about semantics, not syntax / proofs, which require much weaker commitments. Don't want to get into a long discussion in a closed question (that was really a request for further reading anyway). I don't know of a reference devoted to this, but Kunen is a good set theory book that 'cares' about these issues. Also some good discussion elsewhere in the qa-sphere. In addition to some link-following from the duplicate, I would suggest this MO https://mathoverflow.net/questions/300753/circular-or-missing-definition-in-set-theory . – spaceisdarkgreen Dec 20 '18 at 03:50
  • Essentially, if set theory is the language we state all our mathematical ideas in, and we agree on axioms and inference rules for proving things there, then of course set theory subsumes model theory and semantics more generally. See also section 3.1 here for a brief take https://arxiv.org/pdf/0712.1320.pdf . There are some thorny and disputed philosophical questions about what exactly 'the' set theoretical universe is (or if there is such a thing), but a lot of the problems you are wondering about go away when you realize 'models of ZF/C' is different from 'set theory'. – spaceisdarkgreen Dec 20 '18 at 04:24

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