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I am currently learning mathematical logic, and I came across a dilemma. In proving metatheorems (theorems about formal systems), almost all the proofs for said metatheorems used mathematics (induction, set theory, etc). Since most of mathematics is justified by first order logic, which itself is a formal system, wouldn't using mathematical methods to prove metatheorems be circular logic? Not only that, but the proof for the deduction theorem, which itself is an if~then~ proposition, uses the deduction theorem itself. Is this practice natural? I apologize in advance if I am missing something basic or obvious. Thanks for all the help.

Shaun
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Kang Won Lee
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    I don't think most of mathematics is "justified" by FOL; in fact, I would say it is the other way around: we devised FOL precisely to capture certain aspects of our mathematical practice. Since FOL does not have this justificatory purpose, there is no circularity involved. – Nagase May 06 '22 at 18:38
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    You don't need to know the meta theorem to use the formal system. – Alex Kruckman May 06 '22 at 19:03
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    Is it simply that FOL is a machinery for making mathematics rigorous? Also, if that is the case, why would we need proofs for metatheorems like the deduction theorem, and on what basis do we justify proofs about decidability, computability, etc? Thank you so much for the all the comments. – Kang Won Lee May 06 '22 at 19:09
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    Related: https://math.stackexchange.com/questions/173735/how-to-avoid-perceived-circularity-when-defining-a-formal-language, https://math.stackexchange.com/questions/1334678/does-mathematics-become-circular-at-the-bottom-what-is-at-the-bottom-of-mathema, etc. – Hans Lundmark May 06 '22 at 20:21
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    @KangWonLee - I don't personally think FOL is a "machinery for making mathematics rigorous", I think math is already rigorous without FOL. I think FOL is a useful tool for investigating certain structures and certain theories. The fact that a certain theory is formalizable in FOL can gives us a lot of information about the theory, which could be difficult to obtain without the formalization. And we can also show things about theories so formalizable (say, Morley's Categoricity Theorem or Shelah's Main Gap Theorem). – Nagase May 07 '22 at 00:21
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    As for the deduction theorem, it is a useful tool to have, both to prove things inside the system and for proving things about a certain system (I remember it being of help when proving completeness). Note that some systems (say, natural deduction) actually have a kind of "built-in" deduction theorem. Finally, we justify proofs about decidability and computability as we justify proofs about, say, the prime numbers or elliptic curves, by referring the the usual mathematical practice (and axioms, if needed). – Nagase May 07 '22 at 00:23
  • @Nagase: Your last statement is not quite true; see my comment on João Júnior's answer. There is a very big difference between a statement that can be proven in ACA and a statement that can be proven in ZFC but not in ZC. Also, mathematics was not rigorous without FOL; just take a look at the history of sloppy mathematics before Weierstrass gave an FOL-based definition of limits! – user21820 May 24 '22 at 16:49
  • @KangWonLee: You are not missing anything in thinking there is a circularity; I explained that in the linked duplicate. But take note of the comments I have made on this thread; not everything is circular. One philosophically sound approach is to first observe that there appears to be a real-world model of PA−, and from that conclude that there is also a real-world model of ACA (with the set-theory-free interpretation) or at least a very good approximation. This then allows us to work within ACA to prove all the basic facts about FOL. If you want more details, let me know. – user21820 May 24 '22 at 16:55
  • @user21820 - Note that Weierstrass (or Bolzano, for the matter) did not use first-order logic, since it was not invented by his time. Were his definitions not rigorous? Indeed, first-order logic (as opposed to, say, the theory of types) was only isolated as an interesting logic to study at the earliest by Weyl in his Das Kontinuum. Is every mathematics before then not rigorous? Since then, the vast majority of mathematics is also conducted without FOL (in fact, I'd bet most working mathematicians do not even know how to formalize their favorite theories in FOL). Is all of this not rigorous? – Nagase May 24 '22 at 17:13
  • @user21820 - Also, nothing I said was meant to deny that there are differences between a theorem that can be proved in $\mathsf{ACA}$ and one that can be proved in $\mathsf{ZFC}$ or $\mathsf{ZC}$. Reverse mathematics is definitely an interesting area of research, and has many things to say about foundations, but it's not the ultimate arbiter of rigor. – Nagase May 24 '22 at 17:16
  • @Nagase: When I say FOL, I do not mean a modern treatment of it. Rather, I mean the underlying core FOL reasoning. Weierstrass definitely did use FOL. So do most modern mathematicians. People do not have to know a concrete deductive system in order to use FOL. But if they do not understand nested quantifiers and how to reason correctly with them, then that implies a lack of FOL. – user21820 May 24 '22 at 17:17
  • Concerning the difference between theorems of ACA and theorems of ZFC, it was unclear what you meant by "by referring the the usual mathematical practice", and generally people read/misread that to mean that there is a single monolithic "usual practice" with a single axiomatization. I wanted to make clear that the foundation of logic itself does not rest on anything close to "usual mathematical practice", contrary to what many people and even many mathematicians think. – user21820 May 24 '22 at 17:19
  • For instance, I have frequently heard the claim that the semantic-completeness theorem for FOL relies on AC. This claim is extremely misleading, because ACA certainly proves the theorem for countable FOL theories, and that is all that really matters in practice. Why then do people keep repeating the claim about AC? It is because they don't actually know what is needed and why. – user21820 May 24 '22 at 17:22
  • @user21820 - I don't think there is any disagreement between us. First, I was using FOL in the sense of the OP, as a specific (family of) formal system(s), and was then explaining that formalization in this sense is not required for rigor. Since you agree that Weierstrass was rigorous, yet he obviously was not operating with FOL in this sense, we don't disagree. As for the supposed monolithic character of mathematical practice, I agree that it is not monolithic. What I meant is that practicing mathematicians already have standards of rigor without the need for formalization. – Nagase May 24 '22 at 17:24
  • I see. Yes, I don't think we disagree about the mathematics. And perhaps I misread your intent in that statement to which I said "not quite true". However, after talking to you I am reminded of the fact that mathematicians who do not work in logic or set theory very frequently have trouble figuring out whether they are using AC or not, which should not be the case if they had completely rigorous reasoning. So I'm not convinced by your very last claim ("practicing mathematicians already have standards of rigor without the need for formalization"). =) – user21820 May 24 '22 at 17:28
  • @user21820 - to have the standards is not necessarily to enforce them, right? ;) – Nagase May 25 '22 at 18:15
  • @Nagase: Let us continue in chat. =) – user21820 May 25 '22 at 19:24
  • yes, mathematical logic, in the sense of foundation, is bullshit – Masacroso Feb 12 '23 at 20:15

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I understand your discomfort, I also felt the same when I began to study logic. Right on the beginning of propositional logic, one is faced with a combinatorial remark: the number of rows in the truth table of a formula with $n$ propositional variables is $2^n$. A little later, when reading about the language of first-order logic and its semantics, you must know what is an infinite countable set (because so is the alphabet of FOL), what is a function, what is a relation, etc. As far as I understand, this apparent circularity is insurmountable: when you formalize logic, and mathematics, you must know what it is being formalized. This is the reason why one must study logic and set theory in parallel: each one "depends" (in some sense) on the other. But this is not really a circular reasoning, because the formalized theory and the informal knowledge of the concepts being formalized are in different levels (theory vs. metatheory).

João Alves Jr.
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