It's easy to generate all proofs in any system, just pick any axiom, then apply any inference rule, and repeat. So if you ever generate both P and not P, then

It's easy to generate all proofs in any system, just pick any axiom, then apply any inference rule, and repeat. So if you ever generate both P and not P, then you've proven its inconsistent. So if it's inconsistent, it's straight forward to prove. It's only if it's consistent that we can't prove it in any finite time.