Question

My question is about nonstandard analysis, and the diverse possibilities for the choice of the nonstandard model R*. Although one hears talk of the nonstandard reals R*, there are of course many non-isomorphic possibilities for R*. My question is, what kind of structure theorems are there for the isomorphism types of these models?

Background. In nonstandard analysis, one considers the real numbers R, together with whatever structure on the reals is deemed relevant, and constructs a nonstandard version R*, which will have infinitesimal and infinite elements useful for many purposes. In addition, there will be a nonstandard version of whatever structure was placed on the original model. The amazing thing is that there is a Transfer Principle, which states that any first order property about the original structure true in the reals, is also true of the nonstandard reals R* with its structure. In ordinary model-theoretic language, the Transfer Principle is just the assertion that the structure (R,...) is an elementary substructure of the nonstandard reals (R*,...). Let us be generous here, and consider as the standard reals the structure with the reals as the underlying set, and having all possible functions and predicates on R, of every finite arity. (I guess it is also common to consider higher type analogues, where one iterates the power set ω many times, or even ORD many times, but let us leave that alone for now.)

The collection I am interested in is the collection of all possible nontrivial elementary extensions of this structure. Any such extension R* will have the useful infinitesimal and infinite elements that motivate nonstandard analysis. It is an exercise in elementary mathematical logic to find such models R* as ultrapowers or as a consequence of the Compactness theorem in model theory.

Since there will be extensions of any desired cardinality above the continuum, there are many non-isomorphic versions of R*. Even when we consider R* of size continuum, the models arising via ultrapowers will presumably exhibit some saturation properties, whereas it seems we could also construct non-saturated examples.

So my question is: what kind of structure theorems are there for the class of all nonstandard models R*? How many isomorphism types are there for models of size continuum? How much or little of the isomorphism type of a structure is determined by the isomorphism type of the ordered field structure of R*, or even by the order structure of R*?

Answer 1

I think that the nonstandard models of R* will be fairly wild by most reasonable metrics, since the theory is unstable (the universe is linearly ordered). For instance, I don't think that arbitrary models will be determined up to isomorphism by well-founded trees of countable submodels (as they are in ``classifiable'' theories).

EDIT: I'm not sure how many nonisomorphic models there are of cardinality c (the size of the continuum), but there are 2^{2^c} distinct nonisomorphic nonstandard models of theory of R* of size 2^c. A crude counting argument shows that this is the maximum number of nonisomorphic models of size 2^c that any theory with a language of cardinality 2^c could possibly have, which can be considered as evidence that the class of models of the theory of R* is ``wild.''

(This result follows from the proof of Theorem VIII.3.2 of Shelah's Classification Theory, one of his ``many-models'' arguments about unclassifiable theories. In fact, an argument from the second chapter of my thesis applied to this theory shows that you can even build a collection of 2^{2^c} models of size 2^c which are pairwise bi-embeddable but pairwise nonisomorphic.)

It's a good question whether or not you can have two models of this theory which are order-isomorphic but nonisomorphic -- there must be somebody studying o-minimal structures with an answer to this.

Answer 2

Let me offer one counterpoint to John's excellent answer.

Under the Continuum Hypothesis, the ultrapower version of R* will be saturated in any countable language. That is, it will realize all finitely realizable countable types with countably many parameters. Thus, by the usual back-and-forth construction, if we take the reduct to any countable part of the language, such as the ordered field structure and more, then under CH there will be only one saturated model of size continuum.

I'm not sure if John's construction can produce saturated models, but if so, then under CH this observation will answer his question at the end about whether one can have non-isomorphic R*'s that are isomorphic orders or even as ordered fields.

Answer 3

See Robinson's book Non-Standard Analysis (North-Holland 1966)...
Section 3.1 has some remarks about the order type of the non-standard natural numbers.