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This is very much not rigorous, so more of a "fake proof outline" than a "fake proof", but anyway hopefully someone would make explicit the (most important) error in the following line of reasoning:

  • An infinite axiom schema in FOL is semantically equivalent to an infinitary conjunction.
  • The semantics of infinitary logic requires set theory to specify.
  • Ergo infinite axiom schemas are "set theory in sheep's clothing".

Guesses: Is the problem that the above fails to distinguish between arbitrary infinite FOL axiom schemas, and recursively axiomatizable (albeit still infinite) FOL axiom schemas?

Are the semantics of recusively axiomatizable FOL axiom schemas somehow special enough to not require set theory, unlike the semantics of arbitrary infinite collections/"sets" of FOL axioms? Is Craig's theorem relevant?

Maybe the relevant collection of sentences (at least if recursively axiomatizable) can be expressed as a single sentence in $\mathcal{L}_{\omega_1 \omega}$, which is apparently complete (albeit not compact), so there are no semantic dependency issues vis a vis set theory?

Related questions (expressing the idea that infinitary logic is sometimes even more expressive than 2nd order logic):
Is there a specific infinitary sentence second-order logic can't capture?
Is infinitary first-order logic strictly more expressive than weak second-order logic?

hasManyStupidQuestions
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    I don't understand the second step at all. In what sense is it true that one needs set theory to understand the semantics of infinitary conjunctions but one doesn't need set theory to understand the semantics of first-order logic generally? After all, models of first-order theories must be sets, so you need at least a teeny little bit of set theory. – Qiaochu Yuan Jan 08 '23 at 06:49
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    Universal quantifiers are also infinitary conjunctions. So, in what sense are quantifiers not "set theory in sheep clothing"? – Asaf Karagila Jan 08 '23 at 08:45
  • I actually agree with both commenters -- I am also confused to some extent about both of those questions, whether either semantics of FOL or universal quantifiers also require one to choose something like a "platonistic commitment to a completed infinity" or "implicit/background set theory as metatheory". My understanding was that a motivation for formalization is so that we can express everything in finitary, computable symbol/formal languages, because we can get some "empirical confirmation" that those exist e.g. by implementing them on computers. But can that be done with "infinite" axioms? – hasManyStupidQuestions Jan 16 '23 at 15:48

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Any FOL theory with computably enumerable axiom schemas is essentially equivalent (i.e. is bi-interpretable) with another FOL theory with finitely many axioms. You simply encode the (finitely many) rules that underlie the schemas. In some cases, such as finite axiomatization of NBG or ACA0, you can even do this with not much encoding. These clearly demonstrate that the infiniteness of an axiomatization is not at all set-theoretic in itself. Also, second-order logic is way over-hyped if you are concerned about foundations. Second-order theories are important when studying mathematical structures, but not really relevant to true foundations of mathematics.

user21820
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  • These references are very helpful, and this is a good sanity check, thank you! To confirm my understanding, if an infinite axiom schema was not computably enumerable, even if it was "countable", we as humans could never reason about it without access to some (nonexistent) "oracle" capable of "infinite computation", correct? So when textbooks say first order languages have "countably many symbols and countably many axioms" what they really mean in both cases is "computably enumerably many"? Re: SOL, I don't think I actually disagree although explaining fully could potentially be off-topic. – hasManyStupidQuestions Jan 16 '23 at 16:10
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    @hasManyStupidQuestions: Yes, humans cannot do any rigorous reasoning beyond what can be done within a computable formal system. By rigorous we want others to be able to reproducibly verify the reasoning, and the only truly convincing way is via some computable verification procedure. So we cannot use any FOL system that has an uncomputable axiom set, even if it is countable. But logic texts use the term "FOL language" and "FOL theory" without restriction. This is because logic texts are about studying logic, not about using logic. – user21820 Jan 16 '23 at 18:43