# Gödel's second incompleteness theorem and Consistency.

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.

• If you read up on the historical context, that will most likely answer your question. For example, here: plato.stanford.edu/entries/hilbert-program Mar 19, 2023 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. Mar 19, 2023 at 16:07

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).