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In mathematical logic, and particularly in its subfield model theory, a saturated model M is one that realizes as many complete types as may be "reasonably expected" given its size. For example, an ultrapower model of the hyperreals is -saturated, meaning that every descending nested sequence of internal sets has a nonempty intersection.[1]

Definition

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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 strictly less than κ, the model 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 -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]

Motivation

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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 that are not definable (for example, any transcendental element of R is, by definition of the word, not definable in the language of fields). 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 a specific increasing sequence cn 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 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 model M, and the type {x ≠ m : m ∈ M}. 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 that is universally unsatisfied is useless; hence the restriction.

Examples

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Saturated models exist for certain theories and cardinalities:

  • (Q, <)—the set of rational numbers with their usual ordering—is saturated. Intuitively, this is because any type consistent with the theory 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 of real numbers with their usual ordering—is not saturated. For example, take the type (in one variable x) that contains the formula for every natural number n, as well as the formula . This type uses ω different parameters from R. Every finite subset of the type is realized on R by some real x, 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,<) is not ω1-saturated, and not saturated. However, it is ω-saturated, for essentially the same reason as Q—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 a ηα set if and only if it is ℵα-saturated.
  • The countable 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 of Q and the theory of the countable random graph 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 [citation needed] the existence of a proper class of cardinals κ such that κ<κ = κ. The latter identity is equivalent to κ = λ+ = 2λ for some λ, or κ is strongly inaccessible.

Relationship to prime models

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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.

Notes

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References

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Saturated model

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In , a saturated model of a complete theory TT is a that realizes every consistent type over any parameter set of strictly less than the of the model itself. Formally, for an infinite cardinal κ\kappa, a model MM is κ\kappa-saturated if every type in the type space S(A)S(A) over a subset A⊆MA \subseteq M with ∣A∣<κ|A| < \kappa is realized by some element in MM; a model is fully saturated if it is ∣M∣|M|-saturated. This property ensures that saturated models are "rich" in the sense that they contain realizations of all possible consistent extensions of the theory's axioms within the given parameter constraints, making them ideal for studying the theory's without omissions. Saturated models exhibit key structural properties that distinguish them from other models of the same theory. They are universal, meaning every smaller model of TT (of cardinality less than the saturation cardinal) can be elementarily embedded into a saturated one, allowing for extensions that preserve all first-order properties. Additionally, they are strongly homogeneous: any isomorphism between finite substructures extends to an automorphism of the entire model, reflecting a high degree of symmetry. For countable theories with countably many nn-types for each nn, a countable saturated model exists and is unique up to isomorphism, providing a canonical representative for the theory. Examples of saturated models include the field of real numbers R\mathbb{R} as a model of the theory of real closed fields, which is countably saturated and thus realizes all consistent types over finite parameter sets. In general, the existence of saturated models relies on cardinal arithmetic assumptions, such as the generalized continuum hypothesis in some cases, but every model can be elementarily extended to a κ\kappa-saturated one for suitable κ\kappa. These models play a central role in advanced model-theoretic results, such as stability theory and the study of forking independence, by serving as "monster models" that encompass all possible behaviors of the theory.

Definition and Basics

Definition

A model M\mathcal{M} of a complete first-order theory TT in a language LL is said to be κ\kappa-saturated, for an infinite cardinal κ\kappa, if every consistent 1-type over any subset A⊆∣M∣A \subseteq |\mathcal{M}| with ∣A∣<κ|A| < \kappa is realized in M\mathcal{M}. Here, a 1-type p(x)p(x) over AA is a maximal consistent set of formulas in the expanded language L(A)L(A) with one free variable xx, and pp is realized by an element b∈∣M∣b \in |\mathcal{M}| if M⊨ϕ(b)\mathcal{M} \models \phi(b) for every ϕ(x)∈p\phi(x) \in p. The space of all complete 1-types over AA, denoted S1(A)S_1(A), forms a compact Hausdorff topological space known as the Stone space of types over AA, where the basic open sets are defined by principal filters generated by individual formulas. Saturation ensures that M\mathcal{M} realizes every point in S1(A)S_1(A) for small parameter sets AA, providing a rich structure that captures all possible "behaviors" consistent with TT relative to those parameters. For cardinals κ≤λ\kappa \leq \lambda, every λ\lambda-saturated model is κ\kappa-saturated, but the converse does not hold in general; a model may realize all types over sets smaller than κ\kappa without doing so for larger sets up to λ\lambda. This graded notion of saturation allows for precise control over the size and homogeneity of models in the study of TT.

Types and Realization

In model theory, a type over a set of parameters AA is a consistent collection of formulas in a first-order language L\mathcal{L}, all sharing the same finite tuple of free variables, that is maximal with respect to consistency relative to the theory Th(M)\mathrm{Th}(M) of a model MM. More precisely, an nn-type p(xˉ)p(\bar{x}) over AA is a set of LA\mathcal{L}_A-formulas with free variables xˉ=(x1,…,xn)\bar{x} = (x_1, \dots, x_n) such that pp is consistent with ThA(M)\mathrm{Th}_A(M), and it is complete if for every LA\mathcal{L}_A-formula ϕ(xˉ)\phi(\bar{x}), either ϕ∈p\phi \in p or ¬ϕ∈p\neg \phi \in p. The space of all complete nn-types over AA, denoted Sn(A)S_n(A), captures the possible "behaviors" or properties that nn-tuples can exhibit relative to AA. Types are classified by the number of free variables: 1-types concern single elements (n=1n=1), while nn-types address tuples for n>1n > 1. Although saturation involves realizing all nn-types over small parameter sets, the definition is equivalent to realizing all 1-types for infinite cardinals of saturation, as higher nn-types can be reduced to products of 1-types in this context. Thus, the focus in saturation typically lies on 1-types, which describe the complete theory of individual elements over parameters. A tuple bˉ\bar{b} from a model NN (an elementary extension of MM) realizes a type p∈Sn(A)p \in S_n(A) if tp(bˉ/A)=p\mathrm{tp}(\bar{b}/A) = p, meaning N⊨ϕ(bˉ)N \models \phi(\bar{b}) for every ϕ(xˉ)∈p\phi(\bar{x}) \in p. This realization embodies the type by satisfying its entire consistent set of properties simultaneously. In the context of saturation, a model MM is κ\kappa-saturated if every 1-type over any A⊆MA \subseteq M with ∣A∣<κ|A| < \kappa is realized by some element in MM, ensuring that every conceivable consistent behavior relative to small parameter sets is exemplified within the model itself. This property makes saturated models particularly rich, as they embed and extend partial types without omission. For a countable language L\mathcal{L} (i.e., ∣L∣=ℵ0|\mathcal{L}| = \aleph_0) and parameter set AA of cardinality λ≥ℵ0\lambda \geq \aleph_0, the cardinality of the type space satisfies ∣Sn(A)∣≤2λ|S_n(A)| \leq 2^\lambda, since the number of LA\mathcal{L}_A-formulas is at most ℵ0+λ=λ\aleph_0 + \lambda = \lambda, and complete types correspond to maximal consistent subsets, bounded by the power set of the formula space. This bound highlights the potential vastness of possible types, underscoring why saturation requires models of sufficient size to realize them all over small AA.

Properties

Saturation Cardinality

In , the degree of saturation of a model is quantified using infinite cardinals κ, where a model M of a T is defined to be κ-saturated if, for every A ⊆ M with |A| < κ, every complete type over A in the of T is realized by some element in M. This property ensures that M is "complete" in the sense that it omits no consistent extensions of formulas over small parameter sets, making it a robust representative for studying the theory's behavior. A related but distinct notion is that of strong κ-homogeneity, where a model M is strongly κ-homogeneous if every partial elementary between subsets of M of less than κ extends to an of the entire model M. While saturation focuses on type realization, strong homogeneity emphasizes the extendability of isomorphisms, though the two properties often coincide in saturated models of certain theories. The of a κ-saturated model M must satisfy |M| ≥ κ, as the model needs sufficient elements to realize all types over parameter sets approaching κ in size. In stable theories, saturated models achieve a precise balance where the saturation cardinal aligns exactly with the model's , allowing for controlled growth in model size without unnecessary expansions. A model M achieves maximal saturation, often simply called saturated, if it is κ-saturated where κ = |M|, meaning it realizes all types over parameter sets strictly smaller than its own cardinality. For countable models, saturation is particularly restrictive: a countable saturated model exists for a countable theory if and only if the theory is ω-categorical, as only such theories have finitely many types over finite parameter sets, enabling the realization of all of them within countable size.

Homogeneity and Isomorphism

In model theory, a structure MM is said to be homogeneous if every isomorphism between finitely generated substructures of MM extends to an automorphism of the entire structure MM. This property ensures a high degree of symmetry, allowing local isomorphisms to reflect global automorphisms. More generally, a model MM is strongly κ\kappa-homogeneous if every elementary embedding from a subset of MM of cardinality less than κ\kappa into MM extends to an automorphism of MM. Saturation implies strong homogeneity: specifically, a κ\kappa-saturated model of cardinality κ\kappa is strongly κ\kappa-homogeneous. This connection arises because saturation guarantees the realization of all types over parameter sets of size less than κ\kappa, enabling the extension of partial elementary maps via type realization. Thus, saturated models exhibit maximal homogeneity relative to their cardinality, making them particularly useful for studying structural properties in a theory. A key consequence is the isomorphism theorem for saturated models: if TT is a and M\mathcal{M} and N\mathcal{N} are two saturated models of TT of the same κ\kappa, then M≅N\mathcal{M} \cong \mathcal{N}. This uniqueness up to holds because saturation ensures that both models realize exactly the same collection of types, allowing a systematic matching of elements. The proof of this isomorphism typically employs a back-and-forth argument, leveraging the homogeneity of the models. Starting with an empty partial isomorphism, one alternately extends it forward by realizing types in the target model to match elements from the source and backward to ensure surjectivity, using the saturation to guarantee the existence of witnesses for each step without cardinality constraints below κ\kappa. This construction equates the models element by element, confirming their structural identity. While saturated models of the same theory and cardinality are necessarily elementarily equivalent (as they satisfy the same sentences), the converse does not hold; elementary equivalence alone does not imply isomorphism, but saturation strengthens it to ensure identical structure. This distinction highlights how saturation provides a richer form of equivalence beyond mere sentence satisfaction.

Construction

Existence via Löwenheim–Skolem

The upward Löwenheim–Skolem theorem provides a foundational tool for constructing saturated models by enabling the controlled expansion of models while preserving elementary equivalence. Specifically, if TT is a theory in a of μ\mu, and MM is an infinite model of TT, then for any infinite cardinal λ≥max⁡(μ,∣M∣)\lambda \geq \max(\mu, |M|), there exists an elementary extension NN of MM with ∣N∣=λ|N| = \lambda. This theorem, proved using the , allows iterative extensions to larger cardinalities without altering the theory's satisfaction. Saturated models are constructed by iteratively building elementary chains of models to systematically realize types. Starting from any model M0M_0 of TT, one forms a transfinite (Mα)α<δ(M_\alpha)_{\alpha < \delta} of elementary extensions, where at each successor stage Mα+1M_{\alpha+1} realizes all types over MαM_\alpha (using to adjoin witnesses for consistent sets of formulas), and the chain is continuous at limits. The upward ensures each extension can be taken to have at most the next cardinal after the previous stage, yielding a union M=⋃α<δMαM = \bigcup_{\alpha < \delta} M_\alpha that is κ\kappa-saturated for κ≤2∣T∣\kappa \leq 2^{|T|}, provided δ\delta is chosen sufficiently large (e.g., 2∣T∣2^{|T|}). This process achieves saturation by ensuring every type over parameter sets of size less than κ\kappa is realized in MM. A variant of the Henkin construction applies to countable theories, enabling the building of countable saturated models when the theory is ω\omega-categorical. In such cases, the unique (up to isomorphism) countable model realizes all types over finite parameter sets (and is thus countably saturated), as ω\omega-categoricity implies only finitely many nn-types for each nn, hence countably many types overall. Existence of saturated models requires TT to have infinite models, as theories with only finite models lack infinite extensions via Löwenheim–Skolem. For higher saturation levels, specific stability conditions on TT may be needed to bound the number of types sufficiently; without such bounds, full saturation may only be achievable at larger cardinals. If ∣T∣=μ|T| = \mu, then μ\mu-saturated models of cardinality 2μ2^\mu exist.

Uniqueness Conditions

For a complete theory TT, there is at most one saturated model of each κ≥∣T∣\kappa \geq |T| up to . This uniqueness follows from the fact that any two κ\kappa-saturated models of TT of κ\kappa are elementarily equivalent, and their saturation property enables a back-and-forth construction to establish an between them. The existence of saturated models—and thus their uniqueness—in every sufficiently large cardinality κ\kappa requires the theory TT to be or superstable. In unstable theories, such as the theory of dense linear orders without endpoints, saturated models may fail to exist for certain cardinalities, providing a to the presence of unique saturated representatives across all large sizes; for instance, while the rationals Q\mathbb{Q} form the unique countable saturated model, no ℵ1\aleph_1-saturated model exists under the of the . Saturated models serve as universal representatives in the of a theory's models, all other models of the same via elementary embeddings due to their maximality in type realization. In ω\omega-categorical theories, the unique countable model is both saturated and the prime model. In contrast to minimal models, which are atomic and realize only principal types to achieve minimality, saturated models maximize realizations by including elements for every consistent type over parameter sets smaller than their cardinality.

Examples and Applications

Algebraic Structures

In the theory of divisible abelian groups, which is complete and , the κ-saturated model of cardinality κ (for infinite κ) is unique up to and given by the of κ copies of ℚ (the torsion-free part) and, for the torsion part, κ copies of each Prüfer ℤ(p^∞) over all primes p, yielding a structure isomorphic to ℚ^{(κ)} ⊕ \bigoplus_p (\mathbb{Z}(p^\infty))^{(κ)}. This realization ensures that all types over parameter sets of size less than κ are satisfied, reflecting the injective nature of divisible groups as ℤ-modules. For Boolean algebras, consider the theory of atomless Boolean algebras, which admits saturated models. The κ-saturated atomless Boolean algebra of cardinality κ realizes all consistent types over parameter sets smaller than κ. This structure, often constructed via ultrapowers or direct limits, embeds all smaller atomless Boolean algebras and exemplifies saturation by accommodating arbitrary finite partitions and ultrafilter extensions. In the of s over a fixed field F (with including ), the κ-saturated model of cardinality κ is the κ-dimensional over F, where κ is infinite. This model realizes all linear types, such as those specifying over subspaces of less than κ, making it homogeneous and universal for F-vector spaces of that size. The 's superstability ensures such saturated models exist and are unique up to . For the theory ACF_0 of algebraically closed fields of characteristic zero, the κ-saturated model of cardinality κ is the algebraic closure of ℚ adjoining a transcendence basis of size κ. This construction satisfies all types over parameter sets of size less than κ, including those for and roots of polynomials, leveraging the theory's . Morley's categoricity establishes that theories categorical in one uncountable power, such as those of torsion-free divisible abelian groups, vector spaces over ℚ, and algebraically closed fields, have unique saturated models in all larger cardinalities, linking algebraic homogeneity to model-theoretic saturation.

Ordered Fields

The theory of real closed fields (RCF) admits a unique saturated model of each cardinality κ ≥ 2^{\aleph_0}, up to isomorphism. This model is constructed as the real closure of the \mathbb{Q}(t_\alpha \mid \alpha < \kappa), where the t_\alpha are mutually transcendental elements over \mathbb{Q} ordered in such a way that the underlying is the unique \kappa-saturated dense linear without endpoints (of type \eta_\beta where \kappa = \aleph_\beta). The saturation of the model is equivalent to the saturation of its underlying , ensuring that every consistent partial type over a set of parameters of cardinality less than \kappa is realized. In ordered settings, saturation in RCF manifests through the realization of all 1-types over parameter sets of size less than \kappa, which correspond to Dedekind cuts in the ordered structure. Specifically, any Dedekind cut defined by formulas with parameters from the model is filled by an element in the saturated model, reflecting the completeness properties inherent to real closed fields. This realization extends to more complex order configurations, where saturation guarantees that all possible order types consistent with the theory over the parameters are embedded within the model. Dedekind completeness is thus fully attained in the sense that no gaps in the order remain unrealized relative to the parameter set, distinguishing saturated RCF models from non-saturated ones like the real algebraic numbers. A representative example is the saturated real closed field of cardinality continuum, which incorporates the real numbers along with all algebraic reals and transcendentals up to the continuum, embedded within a non-Archimedean structure that realizes infinitesimal and infinite elements. Saturation ensures the realization of all consistent types over parameter sets of cardinality less than the continuum, corresponding to all possible order configurations consistent with the theory. In applications, hyperreal numbers serve as non-standard saturated models of RCF, particularly in non-standard analysis, where their saturation (often countable) allows for the rigorous treatment of infinitesimals and infinite quantities while preserving elementary properties of the reals. This property enables the realization of all possible order types over parameters, facilitating transfer principles and approximations in analysis.

Connections to Other Concepts

Prime Models

In , a prime model of a TT is defined as a model that admits an elementary into every other model of TT. This property positions prime models as minimal structures that serve as "building blocks" for the class of all models of the theory, often being atomic by realizing only isolated types. Prime models relate to saturated models particularly in the context of ω\omega-categorical theories, where the unique countable model up to is both prime and countably saturated. In such theories, the countably saturated model coincides with the prime model because the finite number of types over finite parameter sets ensures that the countable structure realizes all relevant types while embedding elementarily into larger models. A representative example is the of dense linear orders without endpoints, which is ω\omega-categorical; here, the rational numbers Q\mathbb{Q} with the standard order << form the unique countable model, which is both prime and . In contrast to , which can exist at arbitrary cardinalities and realize all types over parameter sets of cardinality less than the model's size, prime models are necessarily countable and minimal in the sense of elementary embeddings. While every prime model is atomic, saturation does not generally imply primeness; this implication holds only in specific settings, such as ω\omega-categorical where the structures overlap. The notion of prime models was introduced by Robert L. Vaught in the early 1960s, with foundational results on their existence and uniqueness for countable complete theories. Connections to categoricity in countable theories were established through works by Erwin Engeler, Jean-Pierre Ressayre, and Michael Morley.

Monster Models

In model theory, a monster model for a complete first-order theory TT is defined as a saturated model of cardinality κ\kappa, where κ\kappa exceeds all relevant parameters such as 2∣T∣2^{|T|}, often chosen as 22∣T∣2^{2^{|T|}} or a strongly inaccessible cardinal to serve as a universal domain encompassing all small models and types of TT. This structure is fully saturated, realizing every complete type over any parameter set of cardinality less than κ\kappa, and strongly homogeneous, meaning that any elementary embedding between small subsets extends to an automorphism of the entire model. The choice of such a large cardinality ensures that the monster model captures the full expressive power of TT without cardinality constraints interfering in typical arguments. The universality of monster models stems from their saturation and homogeneity: every model NN of TT with ∣N∣≤κ|N| \leq \kappa admits an elementary embedding into the monster model M\mathcal{M}, and every type over a parameter set A⊆MA \subseteq \mathcal{M} with ∣A∣<κ|A| < \kappa is realized by some element in M\mathcal{M}. This property positions the monster model as an ambient universe where small substructures can be studied uniformly, with definable sets and types behaving predictably across embeddings. Uniqueness holds up to , as distinct monster models can be elementarily embedded into each other via back-and-forth constructions. Existence of a monster model is established for any TT (assuming no finite models) using the , which yields a κ\kappa-saturated model in ZFC for any cardinal κ\kappa larger than ∣T∣|T|, without requiring the existence of inaccessible cardinals. This construction proceeds by adding constants for types and applying to ensure realizability, resulting in a model that is saturated at the desired level. Monster models are indispensable in stability theory, providing the framework for analyzing forking and independence: in stable theories, non-forking extensions over small sets coincide with independence relations, enabling the decomposition of types into independent components within the monster. They underpin Shelah's classification theory by facilitating bounds on the number of non-isomorphic models in various cardinalities, particularly for theories of finite stability spectrum, where saturation at high cardinals reveals structural invariants. In simple theories, the monster model supports the independence theorem, ensuring that independent types over disjoint sets can be amalgamated without forking.

References

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