Hallucinated Abstract

I actually thought this up around sometime in Feb 2007; I had been reading Sipser’s Intro to Computer Science text, and hallucinated the following abstract while drifting off to sleep:

This paper presents an isomorphism between the set of problems in P and the Natural Numbers, and an isomorphism between the set of problem in NP and the reals. Using this isomorphism the properties of the equivalent classes P and NP are investigated. A property is found which holds for one class but not the other, thus P != NP. The isomorphisms are then used to prove that the Continuum Hypothesis is false because of the relationship between P and NP.

It’s pretty likely that this approach would have a fundamental flaw, but if so, I don’t have enough background to know what it is. Hopefully, I will be able to gain insight on the problem while here at University. Specifically, I’ll begin with the question: What is the cardinality of P?