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In this lengthy thread there's people bickering back and forth about the consistency of PA (Peano Arithmetic) and misunderstandings abound. In reading it I came to an understanding I found useful, though I am not certain of the correctness of. If we take a formalist perspective on the situation and understand mathematical propositions as statements about what happens when we manipulate strings given certain rules, there are two ways of interpreting "the consistency of PA". That is

(a) we can say "PA is consistent" means that any finite series of the applications of logical rules to the axioms of PA will not result in a contradiction, and

(b) the statement "PA is consistent" is rendered as "Con(PA)" appropriately formalized in some formal system (e.g. $\neg (0=1)$ in ZFC) and the question is about provability of "Con(PA)" in the given formal system.

It is my understanding (a) has not/cannot be demonstrated as any proofs done in any given formal system of the consistency of PA depend on the consistency of that formal system, which cannot be proved without resorting to a stronger system, and so on, but (b) has been demonstrated in various formal systems. It seems all disagreement as to whether "is PA consistent" is an open question is the result of one party referring to sense (a) and another referring to sense (b), where for sense (a) it is an open (and perhaps unanswerable) question while in sense (b), which is the usual meaning of an open question, it is definitively closed.

Does this analysis make sense? Am I missing anything important?

Ruggiero Rilievi
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Physical Mathematics
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  • Note for (b), if ¬CON(A) then A⊦CON(PA), also if CON(A), then for A⊦CON(PA) to be true, A need to be the stronger system. – ℋolo Jan 23 '19 at 10:01
  • I have a related question at https://math.stackexchange.com/questions/3084683/legitimacy-of-consistency-proofs – Physical Mathematics Jan 23 '19 at 16:45
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    Your analysis is OK, but I don't think it's a very fruitful topic for MSE for the same reasons as stated in David Roberts's comments on the MO question you link to. – Rob Arthan Jan 23 '19 at 21:38

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Because we know the exact sentence $\text{Con}(\text{PA})$, we know that if (a) was false then (b) would also be false. Any counterexample to (a), if written down, would immediately lead to a counterexample to (b), that is, it would show $\lnot \text{Con}(\text{PA})$.

The deeper issue in threads about "is PA consistent?" is whether that quoted question should be judged by the same standards as other mathematical knowledge, or whether there are somehow stronger standards that should be applied.

By normal mathematical standards of justification, the question "is PA consistent" is settled. But if we raise the burden of proof for that specific question high enough, we can make it impossible to settle.

Carl Mummert
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  • While I agree that if (a) is false then (b) is false, but strangely enough even if (b) is true (by normal standards of truth, that Cons(PA) is provable in some System X), (a) could be false. That is because both systems could be inconsistent so, (a) could be false, that is there is some derivation of an inconsistency in PA, and Cons(PA) could be provable in System X, because if System X is inconsistent, then $\neg$Cons(PA) and Cons(PA) are both provable. That is why I'm not sure if the question "is there a finite set of applications of logical rules to the axioms of PA that lead to a..." cont. – Physical Mathematics Jan 24 '19 at 19:36
  • "contradiction?" is answerable. From a formalist perspective, we have a standard way of interpreting what it means for a given mathematical statement Y to be proven, that is: "In system X, there is a finite set of applications of logical rules to the axioms of System X that result in Y." Now if we ask the above question about the consistency of PA, it does not lend itself to being answered in the negative by some proof of a not provably consistent system. Perhaps then the above question about the consistency of PA is "unmathematical" in some sense. – Physical Mathematics Jan 24 '19 at 19:39
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    When we ask whether a system proves Con(PA), as in (b), the question whether that system also proves ~Con(PA) is somewhat separate. But, as an example, it is not hard to find a system T so that T proves Con(PA) and ZFC proves T is consistent, which means that, by normal mathematical standards, we know "there is a consistent formal system T that proves Con(PA)". If that is what you take (b) to mean. – Carl Mummert Jan 24 '19 at 19:39
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    On the other hand, because the consistency of formal systems that meet the hypotheses of the incompleteness theorems is never provable in those same systems, we will always need to appeal to some "stronger" system to prove the consistency of foundational theories. So if we look at natural foundational systems and we want to prove there is a consistent theory T that proves there is a consistent theory S that proves Con(PA), we can show that is true, but only in a system stronger than PA. We can make the chain of consistency proofs arbitrarily long. – Carl Mummert Jan 24 '19 at 19:42
  • I agree with your comment, but, as I tried to make clear in my second comment you may not have seen yet, I am more concerned with "is there a finite set of applications of logical rules to the axioms of PA that lead to a contradiction?" to which (for what I think are fairly rational reasons) your above example of a proof of Cons(PA) is not a particularly satisfying resolution. Though certainly such a proof goes even beyond the standards of normal math. As such I am becoming more inclined to say my formation of the question is not mathematically answerable. – Physical Mathematics Jan 24 '19 at 19:43
  • We know by normal mathematical standards there are consistent systems that prove Con(PA), and if there was such a string of applications of logical rules as in your last comment, we could prove ~Con(PA) in those same systems, which is impossible by consistency. But really this adds little to the knowledge we already have that PA is consistent - it just pushes the same issues one level higher to metametatheories instead of metatheories. – Carl Mummert Jan 24 '19 at 19:45
  • Let us continue this discussion in chat. – Physical Mathematics Jan 24 '19 at 19:46