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According to Gödel's second incompleteness theorem, no consistent axiomatic system which includes Peano arithmetic can prove its own consistency. As I understand it, this result contributed to spark a crisis in the foundations of mathematics. What I don't really understand is of what use would be proving that a system of axioms is indeed consistent. Indeed, let's assume that we were somehow able to prove that a system of axioms does not produce any contradictions. But in an inconsistent system every statement is true, so we would be able to prove consistency (by contradition). Therefore proving consistency is useful only if the system does not contain contradictions, making the endeavor entirely circular. So we should we care so much about being able to prove the consistency of the axioms? I did not take any course in logic, so I apologize if I misunderstood some results or made wrong assumptions.

PS

I'm aware that this question has been already asked here: Godel's Second Incompleteness and the Assumption of Consistency but I didn't find the answers particularly illuminating.

Alessandro
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  • If you read up on the historical context, that will most likely answer your question. For example, here: https://plato.stanford.edu/entries/hilbert-program/ – Michael Weiss Mar 19 '23 at 15:45
  • Also, Constance Reid’s marvelous biography of Hilbert makes clear what was at stake, and why the result was so upsetting to some people. – Michael Weiss Mar 19 '23 at 16:07

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You're right that a $T$-proof of the consistency of $T$ itself isn't very compelling (although if $T$ purports to be able to resolve all arithmetical questions then via Godel coding this is something $T$ would need to do). However, that's sort of missing the point.

Pre-Godel, the hope was that an "ambitious" theory $T$ could have its consistency proved in a "restrictive" subtheory $S\subset T$ (more precisely, that "infinitistic" mathematics could be proven consistent by "finitistic mathematics," whatever exactly that means). Of course this consistency proof would depend on our acceptance of $S$ itself, but that could still be of value to those who accept $S$ but are skeptical of $T$. The second incompleteness theorem kills this possibility in a very strong way: even if we take $T=S$ itself, we'll still be unable to get what we want (given mild assumptions on $T$ of course).

This is the "negative" force of the theorem, and the sense in which it was shocking at the time it was proved. The "positive" force, so to speak, is in its introduction of the notion of consistency strength, a concept which plays a fundamental role in proof theory and set theory.

Noah Schweber
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  • Does the second incompleteness theorem really kill this hope? It is meaningful to prove that S implies the consistency of T only if we already accept S to be consistent. But if we were able to show that if T is inconsistent, then also S would have to be inconsistent, then, by modus tollens, we could conclude that S is consistent. And it seems to me that this is more or less what happened with ZF and ZFC. – Alessandro Mar 19 '23 at 00:01
  • That is what happened with ZF and ZFC. These theories are equiconsistent. This may have had the effect that some mathematicians became less squeamish about using the AC. (Gregory Moore’s historical study, “Zermelo’s Axiom of Choice”, might answer this question.) But Gödel’s 2nd incompleteness theorem definitely put the kibosh to Hilbert’s Program, even though both Gentzen and Gödel later gives proofs of the consistency of PA. Of course, these used methods going beyond PA; they were not purely finitary. – Michael Weiss Mar 19 '23 at 16:00