TITLE: Is there a well defined "intended" model of the real numbers in the same sense as there is one for the natural numbers?
QUESTION [12 upvotes]: 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?

REPLY [3 votes]: First you need to realize that given a particular model $M$ of a theory $T$, for any statement in the language holds that the statement is either true in $M$ or false in $M$. There are no other possibilities. In that sense, any fixed model of PA or of the real numbers (under whatever theory you want to designate as the theory of the reals) is unambiguous and it decides each and every statement as either true or false. 
Now, things become less clear cut when considering the consequences of the theory $T$ rather than studying a particular model of it. As for PA, as a first order theory it indeed has many nonstandard models. But, if you add to PA the second order statement of induction then it becomes categorical (meaning that every two models of it are isomorphic). A similar phenomenon is true of the real numbers. The first order theory of the reals allows nonstandard models. But, if you add the completeness axiom then it becomes categorical. 
Since most people working with the natural numbers accept the principle of induction (much stronger actually, they accept the axiom of choice) there is essentially just one model of the natural numbers. Since most people working with the reals certainly accept the axiom of completeness (and most also accept the axiom of choice) there is essentially just one model of the reals. 
It should be noted though the sometimes people deliberately consider nonstandard models, like in nonstandard analysis, for the purposes of the study of very ordinary analysis. It has advantages and disadvantages.