If X runs in polynomial time, then there is of course some X' which produces the same outputs on the same inputs, and provably (in PA, I'm pretty sure, certainly in ZFC) runs in polynomial time. So, if there exists an algorithm X which solves 3SAT in polynomial time, then there is an algorithm X' that provably runs in polynomial time, and solves 3SAT. If ZFC cannot prove that X' always produces the right answer, then, I suppose that implies (By Godel's completeness theorem) that there is a model of ZFC in which it doesn't...
uhh... huh.
Does the standard construction of the natural numbers in ZFC, if considered in a non-standard model of ZFC, produce a nonstandard model of PA? That's not what I had expected.
err... I guess there isn't necessarily a thing which is considered "the standard model of ZFC", so I guess I mean, "is there a model of ZFC in which the usual construction of the natural numbers, isn't the standard model of the natural numbers?" (where the model of ZFC is I guess a model within another ZFC or some other set theory, and asking whether the set of natural numbers in that model, is the standard one, is asking about how it compares to the model of the natural numbers in the outer set theory)
uhh... huh.
Does the standard construction of the natural numbers in ZFC, if considered in a non-standard model of ZFC, produce a nonstandard model of PA? That's not what I had expected.
err... I guess there isn't necessarily a thing which is considered "the standard model of ZFC", so I guess I mean, "is there a model of ZFC in which the usual construction of the natural numbers, isn't the standard model of the natural numbers?" (where the model of ZFC is I guess a model within another ZFC or some other set theory, and asking whether the set of natural numbers in that model, is the standard one, is asking about how it compares to the model of the natural numbers in the outer set theory)