Atomic models versus constructible models

Question

I would like to understand better the notion of constructible model. In Rothmaler's Introduction to model theory, a sort of duality is presented between countable saturated models and countable atomic models: the former are homogeneous, universal and unique whereas the latter are homogeneous, prime and unique. The former are hence “thick” models while the latter are “thin” models. Similar statements hold for saturated structures in higher cardinalities, but not for atomic structures in higher cardinalities (in particular, uniqueness might fail). Rothmaler says that the correct generalisation of the notion of atomic to higher cardinalities is the notion of constructible model. However, constructible implies prime. Therefore if the language is countable any constructible model is countable. So there are no constructible models in higher cardinalities, if the language is countable! I don't see how the notion of constructible model is a generalisation of that of atomic model for uncountable cardinalities.

Answer 1

You're correct that if the language is countable, then the notions of constructible model, prime model, and countable atomic model (over the empty set) all agree. Similarly, when $A$ is a countable set of parameters, a model is is constructible over $A$ if and only if it is prime over $A$ if and only if it is countable and atomic over $A$ (these are just constructible/prime/countable atomic models in the countable language obtained by adding the elements of $A$ as constant symbols).

But the sense in which constructible model is a useful generalization of the notion of atomic model to higher cardinalities is in considering uncountable sets of parameters (and hence uncountable languages, when we add these parameters to the language as constant symbols).

Ressayre proved (in an arbitrary language) that if a model $M$ is constructible, then $M$ is prime and atomic, and that constructible models are unique up to isomorphism. But none of the other implications hold: you can have prime models which are not atomic (and hence not constructible), and atomic models which are not prime (and hence not constructible), and prime and atomic models need not be unique up to isomorphism.

In particular, over an uncountable set $A$, atomic models over $A$ don't necessarily have the nice properties we're used to from the countable setting (namely being prime over $A$ and being unique up to isomorphism), while constructible models retain these properties.

The introduction of this paper contains a nice summary of the state of affairs, with references to relevant counterexamples (and you may find the technical results in the paper interesting).