Saturated model

There is an unrelated notion of saturated model in the context of structural equation modeling.

In mathematical logic, and particularly in its subfield model theory, a saturated model M is one which realizes as many complete types as may be "reasonably expected" given its size. For example, an ultrapower model of the hyperreals is countably saturated, meaning that every descending nested sequence of internal sets has a nonempty intersection.

Definition

Let κ be a finite or infinite cardinal number and M a model in some first-order language. Then M is called κ-saturated if for all subsets A ⊆ M of cardinality less than κ, M realizes all complete types over A. The model M is called saturated if it is |M|-saturated where |M| denotes the cardinality of M. That is, it realizes all complete types over sets of parameters of size less than |M|. According to some authors, a model M is called countably saturated if it is $\aleph _{1}$-saturated; that is, it realizes all complete types over countable sets of parameters. According to others, it is countably saturated if it is $\aleph _{0}$-saturated; i.e. realizes all complete types over finite parameter sets.

Motivation

This is a stronger requirement than asking only that M realize all the complete types of the base language; such a model is called weakly saturated (which is the same as being 1-saturated). The difference lies in the fact that M may (and likely does) contain points which are "invisible" to the complete theory of M, in the sense that they are not definable. This argument owes its ultimate success to Gödel's completeness theorem, which guarantees that any consistent additional requirements we can add to the theory of M will be satisfiable by some other model which is logically indistinguishable from M by the original theory. In other words, if we do not explicitly call attention to specific subsets of M, then we miss out on the special features of M which distinguish it from all other elementarily equivalent models, which is often the exact application to which types are put.

The reason it is "reasonable" only to require that M realize types over strictly smaller (in cardinality) subsets is that we can trivially create a type p(x) which takes its parameters from all of M and which cannot be realized in M. Indeed, simply let p(x) be the set of formulas of the form x ≠ a, for arbitrary a in M; then for p(x) to be realized in M, it would have to be satisfied by an element of M unequal to any element of M! Likewise, knowing more about the structure of M we could concoct an unrealizable formula using most but not all the elements of M (for example, in the natural numbers example below we could omit any subset of the formulas so long as the ones which are left still require that x be unbounded above). This would therefore make the notion of saturation useless, so we do not require that such formulas be realized.

Examples

Saturated models exist: for instance, (Q, <) is saturated and the countable random graph is saturated. However, the statement that any model has a saturated elementary extension is not provable in ZFC, since it is equivalent to the generalized continuum hypothesis

Relationship to prime models

The notion of saturated model is dual to the notion of prime model in the following way: let T be a countable theory in a first-order language (that is, a set of mutually consistent sentences in that language) and let P be a prime model of T. Then P admits an elementary embedding into any other model of T. The equivalent notion for saturated models is that any "reasonably small" model of T is elementarily embedded in a saturated model, where "reasonably small" means cardinality no larger than that of the model in which it is to be embedded! Any saturated model is also homogeneous. However, while for countable theories there is a unique prime model, saturated models are necessarily specific to a particular cardinality. Given certain set-theoretic assumptions, saturated models (albeit of very large cardinality) exist for arbitrary theories. For λ-stable theories, saturated models of cardinality λ exist.

References

• Chang, C. C.; Keisler, H. J. Model theory. Third edition. Studies in Logic and the Foundations of Mathematics, 73. North-Holland Publishing Co., Amsterdam, 1990. xvi+650 pp. ISBN 0-444-88054-2
• Marker, David (2002). Model Theory: An Introduction. New York: Springer-Verlag. ISBN 0-387-98760-6
• Poizat, Bruno; Trans: Klein, Moses (2000), A Course in Model Theory, New York: Springer-Verlag. ISBN 0-387-98655-3
• Sacks, Gerald E. (1972), Saturated model theory, W. A. Benjamin, Inc., Reading, Mass., MR0398817