I was pondering the fact that maybe the classical hard complexity-theoretic questions are undecidable, not because they are so themselves, but because some set-theoretic foundations makes the complexity-theoretic foundations shaky.

My thoughts was that perhaps something like the Continuum hypothesis makes P vs NP undecidable. So my question is, is there a "finitary" or otherwise obviously sane environment for complexity theory that would discount this theory immediately? I'm aware of simpler structures where P vs NP has been decided, but I don't know how that would fit in.

I apologize in advance if this doesn't make sense.