Background: We know that PA has more models than the intended model, N, because it is not strong enough and is also satisfied by non-intended models, known as non-standard models of arithmetic. When we talk about the standard model N, I somehow assume that there is some way to characterize it without ambiguity, so it is well defined and can be identified as the only model that is isomorphic to the natural numbers that we use everyday. Every statement of PA has a specific truth value on N (either true or false, regardless of our ability to know the answer). But I am not sure if that is also the case for the real numbers. Informally, they are a value that represents a quantity along a continuous line. Also, they can be defined axiomatically up to an isomorphism in different ways. They are also been shown to "fill" the real line, so there are no more numbers than them on it.
Question(s): Each statement about the naturals has a specific truth value: Is this the same case for the reals? are they defined with such precision? My doubt comes from the fact that there are models of set theory in which the CH is true and others in which it is false. Do this mean that the actual reals do not have a specific truth value for the CH, or does it mean that they have a specific value (which we don't know yet), and that models with a different value are models of non-standard reals?