As I understand it, it is possible to prove the consistency of a given axiomatic system using a stronger axiomatic system, but no system can be proven to be absolutely consistent (essentially, the consistency of the given axiomatic system is contingent upon the consistency of the stronger axiomatic system; the system is consistent iff the stronger system is consistent).
Is there a way to conclusively prove the inconsistency of an axiomatic system, apart from simply chancing upon a contradiction as Russell did?
Side notes:
- I'm assuming the answer is No, since if it were possible to determine if a system is inconsistent, it would be possible to prove if a given statement is decidable in an axiomatic system (this can be done by constructing two modified axiomatic systems - one in which the given statement is true, and another where it is false - and checking which one in inconsistent)
- A related, slightly naive, question: Is it possible to prove the inconsistency of a stronger axiomatic system by detecting a contradiction in a weaker axiomatic system?
If a system is inconsistent (which can be shown by deriving a single contradiction), then of course every system containing this system is inconsistent either.
But a stronger system can be inconsistent even if the weaker system is consistent. Just add a contradictionary axiom as $\ 0=1\ $ to the system.
I do not think that we could show that , for example , PA is inconsistent without detecting a contradiction.