I have recently begun curious in set theory, and when I researched this subject I saw that all axiomatizations of set theory, such as ZFC and NBG, are expressed in the language of first order logic. However, when I began reading any book that explains formal logic, they used the notion of a set, such as the set of symbols and variables. This looks much like circular reasoning. It is possible to introduce propositional logic, first order logic and higher order logic without formally appealing to the notion of a set? Or is this too much of an intuitive concept that we can’t work around it? And if yes, I would gladly appreciate any recommendations of books that treat formal logic without using sets, or any set theoretical results, as I am currently trying to understand the axiomatization of ZFC.
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3Some discussions on this subject over on [math.se]: https://math.stackexchange.com/q/121128/622 https://math.stackexchange.com/q/1334678/622 https://math.stackexchange.com/q/843685/622 – Asaf Karagila Jun 09 '23 at 12:18
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1Some related posts on [math.se]: An (apparently) vicious circle in logic, An (apparently) vicious circle in logic and When does the set enter set theory? – Martin Sleziak Jun 10 '23 at 10:14
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2On MO, I was able to find these posts, which are related at least to some extent: The egg and the chicken, Do set-theorists use informal set theory as their meta-theory when talking about models of ZFC?, Don't the axioms of set theory implicitly assume numbers?, The sets in mathematical logic and Are there textbooks on logic where the references to set theory appear only after the construction of set theory? – Martin Sleziak Jun 10 '23 at 10:14
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Like many nominalist enterprises, this can be done in principle. In practice, things get very tedious and long-winded very quickly. Compare this to the situation of mathematical rigour in physics: many non-rigorous arguments could be made completely mathematically rigorous, but 1) the (very important) physics intuition would often get lost, and 2) the benefits are what? The same applies here, I would say. – Sam Sanders Jun 12 '23 at 07:29
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Besides the post on circularity that Asaf linked to, see also this post on building blocks. Essentially, when you want to set up FOL for actual concrete formal systems, and reason about it, you need only ACA, which supports reasoning about ℕ and subsets of ℕ that have an arithmetical defining formula. You do not need actually set-theoretic assumptions. – user21820 Mar 11 '24 at 14:54