Pretty sure this can’t happen. We stipulated that we had the shortest proof for P. How did P and P’ get proved without proving P first? Or alternatively, without there being a shorter proof available for P without P’?
What if P = “The prime numbers are infinite or there are exactly 1337 of them” and P’ = “There are not exactly 1337 prime numbers.”
The shortest way to prove P involves proving (P and P’), which is the statement “The prime numbers are infinite.” It would take a roundabout argument indeed to exclude all finite cardinalities besides 1337, without also excluding 1337 and accidentally proving (P and P’).
I don’t have an example to hand. I just have a feeling that there might be some case where the shortest proof of (P and P’) is a component of the shortest proof of P, rather than the other way around.
I don’t think this necessarily follows. Suppose the last few lines of the proof of P read something like:
Pretty sure this can’t happen. We stipulated that we had the shortest proof for P. How did P and P’ get proved without proving P first? Or alternatively, without there being a shorter proof available for P without P’?
What if P = “The prime numbers are infinite or there are exactly 1337 of them” and P’ = “There are not exactly 1337 prime numbers.”
The shortest way to prove P involves proving (P and P’), which is the statement “The prime numbers are infinite.” It would take a roundabout argument indeed to exclude all finite cardinalities besides 1337, without also excluding 1337 and accidentally proving (P and P’).
I don’t have an example to hand. I just have a feeling that there might be some case where the shortest proof of (P and P’) is a component of the shortest proof of P, rather than the other way around.