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The usual semantics of first-order theories involves models who have a structure-set. Hence it is required to use some sort of set theory as the metatheory. However it is also widely known that one can do logic with a minimal metatheory, say a fragment of Peano. If we are using such a metatheory, what will the semantics become ? What will replace the notion of "set-model" ?

C. Dubussy
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    I was under the (uneducated) impression that metatheories had to be "stronger" than the theories we want to work with. This question makes me think this impression might be wrong. Is it? Titimathy. – Uretki May 03 '23 at 19:37
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    @Uretki that impression is wrong, in a sense: you can have very weak meta-theories if you like, though if you want to prove consistency of the object theory, you do need to be "stronger" in some sense. However that does not necessarily mean "have sets", it just means "can prove more things" (in some sense). – cody May 03 '23 at 21:13
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    @Uretki Even weak theories can still prove conditional statements about strong systems. E.g. "If ZFC is consistent then ZFC can't prove that ZFC is consistent" is provable in PA, even though PA is vastly weaker than ZFC. – Noah Schweber May 04 '23 at 03:02

2 Answers2

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Essentially, "model" in the usual set-theoretic sense gets replaced by "complete theory with the strong witness property" (for some subtleties around this point, and in particular the reason why complete theories per se aren't enough, see these old MSE answers of mine: 1, 2). It turns out that every "definable" consistent set of sentences has a "definable" extension to such a theory, so we see enough such theories to recover the usual notions of satisfiability and entailment. The key result here is the arithmetized completeness theorem, which is an analogue of the usual set-theoretic completeness theorem inside first-order arithmetic. Sadly I'm not aware of a great introduction to it, but see Dean, Bernays and the completeness theorem.

Noah Schweber
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If you don't mind a tiny bit of set-theoretic notions, you can do a lot with ACA0 (see this PDF), which is conservative over PA. If you want even weaker systems than that, you might look at WKL0, which is necessary to get you the semantic-completeness theorem for countable FOL theories.

Note that even with ACA0, one can argue that there is no true reliance on a notion of sets, since we can take arithmetical sets as our intended model of ACA0, and hence interpret a subset of ℕ to be simply a property on ℕ given by an explicit 1-parameter sentence over PA. For example, the semantic-completeness theorem states that "every consistent FOL theory has a model". In this interpretation it means "every FOL theory represented by an arithmetical property on ℕ that is consistent has a model represented by some arithmetical property on ℕ (with suitable coding)". And this does apply to every practical FOL theory including ZFC.

user21820
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