What exactly are you denying when you deny that the continuum hypothesis has a definite truth value? After all it's very easy to prove "CH is either true or false" in whatever formal system you prefer, with some notable but unpopular exceptions.
I'm not completely sure of that myself, but consider this analogy. Let PA+X be a formal system that consists of the axioms of PA plus a new axiom that introduces a new symbol X and simply says "X is an integer", without saying anything more about X. Then it's easy to prove "X is either even or odd" in PA+X, but it would be wrong to say that PA+X has a unique distinguished "standard model" that pins down the parity of X. So my statement about CH is more of a statement about our intuitions possibly misfiring when they say a formal system must have a unique standard model.
With thanks to Paul Christiano
My previous post left one important issue unresolved. Second order logic needed to make use of set theory in order to work its magic, pin down a single copy of the reals and natural numbers, and so on. But set theory is a first order theory, with all the problems that this brings - multiple models, of uncontrollable sizes. How can these two facts be reconciled?
Quite simply, it turns out: for any given model of set theory, the uniqueness proofs still work. Hence the proper statement is:
Often, different models of set theory will have the same model of the reals inside them; but not always. Countable models of set theory, for instance, will have a countable model of the reals. So models of the reals can be divided into three categories:
And similarly for the natural numbers.