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There are many examples in the literature (for example, in this question) where the author says that something "is provable in Peano arithmetic" or "is not provable in Peano arithmetic", and I am trying to understand what this means.

From what I have read so far, if T is a sentence of Peano arithmetic, it seems that saying "T is provable in Peano Arithmetic" means "T can be derived from the axioms of Peano Arithmetic by applying inference rules".

But what are these inference rules? Almost all sources that say something like "X is provable in Peano arithmetic" do not specify this. Among those that do, they seem to refer to different inference rules. For example, these notes seem to suggest that LK proofs are used. The book The Logic of Provability uses instead two inference rules: modus ponens, and generalisation from universalisation.

Hence my question: is there any particular set of inference rules that everyone generally understands to be "the inference rules of Peano arithmetic"? Or is it the case that it does not matter which inference rules you use, as long as they satisfy some properties? If so, what are these properties?

niilogunay
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    The inference rules are those of the "underlying logic". If with Peano arithmetic we refer to the first-order formal theory of arithmetic, the rules are those of first.order logic. See First-order logic. – Mauro ALLEGRANZA Apr 07 '22 at 13:24
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    As you can see from Wiki's entry, there are many proof systems (axioms+rules) but all are "equivalent" in the sense that they are all sound and complete wrt classical semantics. – Mauro ALLEGRANZA Apr 07 '22 at 13:33
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    I would like to amplify Mauro's comment so that you don't miss it. Sometimes this sort of question has no clear answer, or multiple answers. But here the answer is clear: you want to look up “First-order logic”, as Mauro said. – MJD Apr 07 '22 at 13:52
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    Having read the Wiki entry, I take it, then, that the answer is, roughly, "any proof system that is both sound and complete with respect to first-order logic." For example, LK proofs or Natural Deduction. That makes sense.

    Thank you @MauroALLEGRANZA and MJD, this answers my question.

     – niilogunay Apr 07 '22 at 14:54
  • Related question. I did not see it when I searched the site before asking this question, nor did it appear in the list of "related questions" suggested by the site when I was writing my question, but it appeared just now in my sidebar. – niilogunay Apr 07 '22 at 15:25
  • @MauroALLEGRANZA That shoves off the question to "what are the rules of inference of first-order logic?" And of course, there's not a unique answer, as your second comment suggests. – Doug Spoonwood Apr 07 '22 at 17:24
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    @niilogunay: The duplicate I linked you to gives one deductive system for (many-sorted) FOL, which is in my opinion one of the most user-friendly systems. PA is an FOL theory, meaning that a proof over PA obeys the deductive rules for FOL but may at any point use an axiom of PA. The section "Peano Arithmetic" gives one possible axiomatization of PA over FOL, which not only is practically usable but also has nice theoretical properties. – user21820 Apr 13 '22 at 21:47
  • @user21820: Thank you for the link. One thing that doesn't seem to be explicitly said in the duplicate, however, but it's necessary to answer the question, is that one can use any proof system for PA as long as it is sound and complete for FOL, as MauroALLEGRANZA points out in the second comment. – niilogunay Apr 14 '22 at 23:13
  • @user21820 for example, your comment says "the deductive rules for FOL". This is an example of the kind of wording that was originally confusing for me, which led me to ask the question. Now I know there is no unique set of such rules :) – niilogunay Apr 14 '22 at 23:21
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    @niilogunay: Yes, that's correct. Actually, that point is mentioned in the duplicate, but in simpler terms: "Each inference rule is chosen to be sound, [...]. We say that these rules are truth-preserving. If you choose carefully enough, you can make it so that the rules are not just truth-preserving but also allow you to deduce every (well-formed) statement that is necessarily true (in all situations)." And in case you are wondering, although the axioms technically don't matter as long as the theorems are the same, logicians generally think of PA as PA− plus induction. – user21820 Apr 15 '22 at 05:17
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    So when we think about a theorem of PA, we often think about how much induction is needed to prove it, which is not well-defined if we have no specific axiomatization in mind. It is also why we adopt PA− as the base theory rather than Peano's axioms minus induction, because we do need the discrete ordered semi-ring structure to do quite a lot of things in the absence of induction. Of course, this is just how we think; we would still state results precisely by saying something like "( PA− + Σ[1]-induction ) proves Q but PA− does not.". – user21820 Apr 15 '22 at 05:23

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