For a frame of reference, I'm an undergraduate in mathematics who has taken the introductory analysis series and the graduate level algebra sequence at my university.
In my analysis class, our book lists axioms describing the structure of the reals. This seemed unnatural to me, as we can often write definitions or lists of rules that no set actually satisfies. So the axiomatic approach doesn't reassure me that the thing we are discussing can actually exist.
Our teacher talked to us about dedekind cuts as a way of explicitly constructing reals, which seemed more useful to me.
In my modern algebra class we discussed completion of the rationals as a way to construct the real numbers.
But this leads me to wonder – How do we know that all of these approaches and constructions result in the same structure, namely $\mathbb{R}$? Also, are the real numbers the only complete, totally ordered field? If so, why?
Best Answer
To paint a more complete picture, you are right in that an axiomatization may very well have no model. An axiomatization is meaningless if nothing satisfies it. But if we can prove that there is a model, and the axioms are the only properties we care about, then we can happily work within the axiomatization to prove theorems about the structure that we have axiomatized. That is precisely what we are doing when we use an axiomatization of the real numbers to prove theorems about real numbers.
Now there are two common axiomatizations of the reals. One is a second-order theory including the completeness axiom (every subset of the reals with an upper bound has a least upper bound). By itself, this axiom is useless, because there is no axiom that asserts the existence of any set of reals at all! However, when we use this second-order theory we are always working outside the theory in the foundational system (such as ZFC set theory) where we do have axioms that allow construction of subsets of the reals. It is this particular axiomatization that has a unique model; there is a unique model of the second-order theory of the reals up to isomorphism. That implies immediately that all structures that you construct (like the Cauchy completion of the rationals or the Dedekind completion of the rationals) that satisfies this second-order axiomatization must be isomorphic to one another.
The core reason behind the uniqueness of complete ordered field is that any two such fields must contain an isomorphic copy of the rationals, and every element in each field cuts the rationals in the field into two parts, and the lower part has a least upper bound, and that different elements cut the rationals in different ways (by the Archimedean property that follows from the completeness property as well). This gives a one-to-one correspondence between the reals in one field to the reals in the other field. One can say that it is the rigidity and denseness of the rationals that is the key.
However, the other common axiomatization of the reals is the theory of real closed fields. Note that this axiomatization does not have a unique model. The computable reals form a countable real closed field, and satisfies every first-order sentence that $\mathbb{R}$ satisfies. It may be instructive to see the same phenomenon with the natural numbers, which form the unique model of the second-order Peano's axioms (which was his original formulation) up to isomorphism, while there are many non-isomorphic models of first-order PA. The distinction here between the first-order induction schema and the single second-order induction axiom must be appreciated for one to understand how the second-order theory can pin down the natural numbers unlike the first-order one. More specifically, second-order induction applies to every subset of the natural numbers (as seen from the foundational system), while first-order induction applies only to subsets that can be described using an arithmetical formula. There are uncountably many subsets, but only countably many formulae.