Saturated model
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In mathematical logic, and in particular model theory, a saturated model M is one which realizes as many complete types as may be "reasonably expected" given its size.
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Definition
Let κ be an infinite cardinal number and M an infinite model in some countable first-order language (whose nature is unimportant to the definition). Then M is κ-saturated if and only if for all subsets A ⊆ M of cardinality less than κ, M realizes all complete types over A. M is saturated if and only if it is |M|-saturated; that is, it realizes all complete types over sets of parameters of size less than |M|.
Motivation
This is a stronger requirement than simply asking that M realize simply all the complete types of the base language; such a model is called weakly 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 to attention 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 that is impossible to satisfy. Indeed, simply let p(x) be the set of formulas of the form x ≠ a, for arbitrary a in M; then p(x) is necessarily satisfied by an element of M unequal to eny 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. The natural numbers (N, S) (where S is the successor function, S(n) = n + 1 for any integer n) are not saturated: the type p(x) containing
- { x > 0, x > S(0), x > S(S(0)), …, x > S(… (S(0)) … ), … }
is not realized (since it would necessarily be realized by an infinite number!).
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 (can someone confirm whether this holds in uncountable languages as well?) there is a unique prime model, saturated models are necessarily specific to a particular cardinality. One generally talks about countable saturated models of countable theories (in countable languages) as a result, as the Löwenheim-Skolem theorem guarantees that any theory in a countable language has a countable model and thus there is already a rich set of examples in this restricted domain of research.
References
- Marker, David (2002). Model Theory: An Introduction. New York: Springer-Verlag. ISBN 0-387-98760-6