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Gödel's second incompleteness theorem and Consistency.

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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.


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.



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