This article includes alist of references,related reading, orexternal links,but its sources remain unclear because it lacksinline citations. Please helpimprove this article byintroducing more precise citations.(January 2010) (Learn how and when to remove this message)

Inmathematical logic, and particularly in its subfieldmodel theory, asaturated modelM is one that realizes as manycomplete types as may be "reasonably expected" given its size. For example, anultrapower model of thehyperreals is${\displaystyle {\mathrm{\aleph }}_1}$-saturated, meaning that every descending nested sequence ofinternal sets has a nonempty intersection.^[1]

Letκ be afinite orinfinitecardinal number andM a model in somefirst-order language. ThenM is calledκ-saturated if for all subsetsA ⊆M ofcardinality strictly less thanκ, the modelM realizes allcomplete types overA. The modelM is calledsaturated if it is |M|-saturated where |M| denotes the cardinality ofM. That is, it realizes all complete types over sets of parameters of size less than |M|. According to some authors, a modelM is calledcountably saturated if it is${\displaystyle {\mathrm{\aleph }}_1}$-saturated; that is, it realizes all complete types over countable sets of parameters.^[2] According to others, it is countably saturated if it is countable and saturated.^[3]

The seemingly more intuitive notion—that all complete types of the language are realized—turns out to be too weak (and is appropriately namedweak saturation, which is the same as 1-saturation). The difference lies in the fact that many structures contain elements that are not definable (for example, anytranscendental element ofR is, by definition of the word, not definable in the language offields). 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 that we may otherwise miss—for example, a bound on aspecific increasing sequencec_n can be expressed as realizing the type{x ≥c_n :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 ℵ₁-saturated structure will.

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

Saturated models exist for certain theories and cardinalities:

• (Q, <)—the set ofrational numbers with their usual ordering—is saturated. Intuitively, this is because any type consistent with thetheory is implied by the order type; that is, the order the variables come in tells you everything there is to know about their role in the structure.
• (R, <)—the set ofreal numbers with their usual ordering—isnot saturated. For example, take the type (in one variablex) that contains the formula${\displaystyle x>-\frac{1}{n}}$ for every natural numbern, as well as the formula. This type uses ω different parameters fromR. Every finite subset of the type is realized onR by some realx, so by compactness the type is consistent with the structure, but it is not realized, as that would imply an upper bound to the sequence −1/n that is less than 0 (its least upper bound). Thus (R,<) isnot ω₁-saturated, and not saturated. However, itis ω-saturated, for essentially the same reason asQ—every finite type is given by the order type, which if consistent, is always realized, because of the density of the order.
• A dense totally ordered set without endpoints is aη_α set if and only if it is ℵ_α-saturated.
• Thecountable random graph, with the only non-logical symbol being the edge existence relation, is also saturated, because any complete type is isolated (implied) by the finite subgraph consisting of the variables and parameters used to define the type.

Both the theory ofQ and the theory of the countable random graph can be shown to beω-categorical through theback-and-forth method. This can be generalized as follows: the unique model of cardinalityκ of a countableκ-categorical theory is saturated.

However, the statement that every model has a saturatedelementary extension is not provable inZFC. In fact, this statement is equivalent to^{[citation needed]} the existence of a proper class of cardinalsκ such thatκ^<κ = κ. The latter identity is equivalent toκ =λ⁺ = 2^λ for someλ, orκ isstrongly inaccessible.

The notion of saturated model is dual to the notion ofprime model in the following way: letT be a countable theory in a first-order language (that is, a set of mutually consistent sentences in that language) and letP be a prime model ofT. ThenP admits anelementary embedding into any other model ofT. The equivalent notion for saturated models is that any "reasonably small" model ofT 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 alsohomogeneous. 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. Forλ-stable theories, saturated models of cardinalityλ exist.

• 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
• R. Goldblatt (1998). Lectures on the hyperreals. An introduction to nonstandard analysis. Springer.
• Marker, David (2002).Model Theory: An Introduction. New York: Springer-Verlag.ISBN 0-387-98760-6
• Poizat, Bruno; (translation: 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.,MR 0398817

• Mathematical logic
• Model theory
• Nonstandard analysis

• Articles with short description
• Short description is different from Wikidata
• Articles lacking in-text citations from January 2010
• All articles lacking in-text citations
• All articles with unsourced statements
• Articles with unsourced statements from July 2018