# Saturated model

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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 $\aleph _{1}$ -saturated, meaning that every descending nested sequence of internal sets has a nonempty intersection, see Goldblatt (1998).

## 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 AM 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

{{ safesubst:#invoke:Unsubst||$N=Technical |date=__DATE__ |$B= {{#invoke:Message box|ambox}} }} The seemingly more intuitive notion – that all complete types of the language are realized – turns out to be too weak (and is, appropriately, named weak saturation, which is the same as 1-saturation). The difference lies in the fact that many structures contain elements which are not definable (for example, any transcendental element of R is, by definition of the word, not definable in the field language). However, they still form a part of the structure, so we need types to describe relationships with them. Thus we allow sets of parameters from the structure in our definition of types. This argument allows us to discuss specific features of the model which we may otherwise miss – for example, a specific increasing sequence cn having a bound can be expressed as realizing the type {x > cn : n ∈ ω}, which uses countably many parameters. If the sequence is not definable, this fact about the structure cannot be described using the base language, so a weakly saturated structure may not bound the sequence, while an ω-saturated structure will.

The reason we only require parameter sets which are strictly smaller than the model is trivial: without this restriction, no infinite model is saturated. Consider a model M, and the type {xm : mM}. Each finite subset of this type is realized in the (infinite) model M, so by compactness it is consistent with M, but is trivially not realized. Any definition which is universally unsatisfied is useless; hence the restriction.

## Examples

Saturated models exist for certain theories and cardinalities:

Both of these theories can be shown to be ω-categorical through the back-and-forth method. This can be generalized as follows: the unique model of cardinality κ of a countable κ-categorical theory is saturated.

However, the statement that every model has a saturated elementary extension is not provable in ZFC. In fact, this statement is equivalent to the existence of a proper class of cardinals κ such that κ = κ. The latter identity implies that either κ = λ+ = 2λ for some λ, or κ is weakly inaccessible.

## 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.