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If set theory is based on logic (being a first-order theory with equality), but such theories assume models which require a domain, which is a set, how is there no circularity in this construction?

Edit

Forget about models, even a formal theory (in terms of which a language is defined, in terms of which interpretations are defined, etc.) immediately uses the notions of sets and subsets: “there is a set of symbols, a set of expressions with a subset of well-formed formulas”.

Jos van Nieuwman
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    As a starting place, Model Theory $\neq$ logic. – Joe Nov 17 '22 at 01:10
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    You don’t need to know anything about models to write down the rules of first-order logic. Models are a way of studying the properties of logic. – Mark Saving Nov 17 '22 at 01:18
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    At some point there has to be some level of circularity indeed. But this is more of an issue of formality than anything, as miraculously we somehow agree on what these symbols do. – Graviton Nov 17 '22 at 07:36
  • @Joe you’re right, that was sloppy. I’ve corrected it, but also I feel like the answer is now indeed clear to me, however unsatisfactory.. :/ – Jos van Nieuwman Nov 22 '22 at 15:32

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