Is there a well defined "intended" model of the real numbers in the same sense as there is one for the natural numbers?

Question

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?

Answer 1

The real numbers, much like the natural numbers, have a canonical second-order model. This means that given a model of set theory there is just one model up to isomorphism.

Of course this model can change if we change the model of set theory, but so can the standard model of the natural numbers. That is if $M$ and $N$ are different models of ZFC they might have different collections of what they perceive as natural numbers or as real numbers.

When we talk about the real numbers as a first-order theory we often consider the theory of real closed fields. We can consider models of this theory which are non-standard. For example the hyperreal field is an example of such model, and non-standard models are useful for non-standard analysis. What makes them non-standard? Well, they usually have "infinite" numbers, which are numbers that are larger than any finite repetition of adding $1$ to itself, much like how non-standard integers exist in non-standard models of PA.

However CH has nothing to do with this. Because CH is not something we can really formulate in the language of ordered fields. In this theory we can't really say one set has a larger cardinality than another. This would be equivalent to asking whether or not there are non-standard models of the real numbers because there are non-abelian groups, and so being an abelian group is independent of group theory (which the real numbers are a model of, of course).

It is true that within a fixed model of ZFC the standard model of the natural numbers is always countable; but it is also true that the real numbers always have cardinality equal to the power set of the standard model of the natural numbers, that is, $|\mathbb R|=2^{\aleph_0}$, regardless to it being $\aleph_1$ or $\aleph_{50043}$. In fact this is true even if the axiom of choice fails and $2^{\aleph_0}$ is not an ordinal at all.

Answer 2

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.