Types of Models

Why This Matters

In Model Theory, understanding different types of models isn't just about memorizing definitions—it's about grasping how models relate to their theories and what structural properties make certain models special. You're being tested on your ability to recognize when a model serves as a foundation for others, when it's "rich enough" to realize all possible behaviors, and when it exhibits the kind of symmetry that makes mathematical analysis tractable. These concepts—saturation, homogeneity, primality, stability—form the backbone of modern model-theoretic classification.

The models you'll encounter here fall into distinct conceptual families: some describe how much a model can realize, others describe embedding and foundational relationships, and still others characterize structural rigidity and definability. When you see a question about types of models, don't just recall the definition—ask yourself what role that model plays in the broader theory. Does it serve as a universal container? A minimal building block? A perfectly symmetric structure? That conceptual understanding is what separates strong answers from mediocre ones.

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Models Defined by Realization of Types

These models are characterized by how many types they can realize—essentially, how "rich" or "complete" they are in accommodating possible behaviors consistent with the theory.

Saturated Models

• Realizes all types over parameter sets up to a given cardinality—a $\kappa$-saturated model realizes every type over any set of parameters of size less than $\kappa$
• Richness measure indicating the model can accommodate virtually any consistent local behavior; the more saturated, the fewer "gaps" in what the model contains
• Central to classification theory because saturated models serve as canonical representatives for studying a theory's structural properties

Atomic Models

• Every element satisfies an isolated type—each tuple in the model is "pinned down" by a single formula that generates its complete type
• Minimal realization property in the sense that atomic models realize only the types that must be realized; the opposite extreme from saturation
• Existence tied to countable complete theories with the property that isolated types are dense in the type space

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Models Defined by Embedding Relationships

These models are characterized by how they relate to other models through elementary embeddings—they serve as foundational structures or universal containers.

Prime Models

• Elementarily embeds into every model of the theory—if $M$ is prime for theory $T$, then $M \preccurlyeq N$ for any $N \models T$
• Unique up to isomorphism for complete theories over countable languages; serves as the canonical "smallest" model
• Coincides with atomic models in the countable case—a countable model is prime if and only if it is atomic

Universal Models

• Every model of bounded size embeds into it—a $\kappa$-universal model contains an isomorphic copy of every model of cardinality at most $\kappa$
• Existence requires set-theoretic conditions such as GCH or saturation assumptions; not guaranteed for arbitrary theories
• Dual to prime models in the embedding hierarchy—prime models embed into everything, universal models have everything embed into them

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Models Defined by Symmetry and Homogeneity

These models exhibit strong internal symmetry—what's true locally can be extended globally, making the model "look the same" from every vantage point.

Homogeneous Models

• Partial isomorphisms extend to automorphisms—any isomorphism between finite (or small) substructures extends to an automorphism of the entire model
• Uniform local-to-global structure ensuring the model has no "special points" that break symmetry; every finite configuration appears in the same global context
• Combines with saturation to yield powerful classification tools; saturated models of complete theories are always homogeneous

Complete Models

• Decides every sentence in the language—for every sentence $\varphi$, either $M \models \varphi$ or $M \models \neg\varphi$
• Associated with complete theories where the theory $\text{Th}(M)$ is maximally consistent; note: every model is complete in this sense as a structure
• Terminology caution: "complete model" sometimes refers to models of complete theories rather than a special model property

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Models Defined by Closure Properties

These models are characterized by closure under certain operations or formulas—they contain everything that "should" exist given what's already present.

Existentially Closed Models

• Realizes witnesses for existential formulas—if $M \models \exists x \, \varphi(x, \bar{a})$ for parameters $\bar{a} \in M$, then some $b \in M$ satisfies $\varphi(b, \bar{a})$
• Model-theoretic analogue of algebraic closure for fields; existentially closed fields are exactly algebraically closed fields
• Key to model companions since the theory of existentially closed models (when it exists) forms the model companion of the original theory

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Models Defined by Structural Rigidity

These models exhibit strong constraints on definable sets or type counts, leading to highly regular and classifiable behavior.

Strongly Minimal Models

• Every definable set is finite or cofinite—in a strongly minimal structure, definable subsets of the universe have the simplest possible complexity
• Induces a pregeometry via algebraic closure, allowing dimension theory analogous to vector spaces; this is the foundation of geometric stability theory
• Canonical examples include algebraically closed fields, vector spaces, and the theory of equality

Stable Models

• Bounded type counts over parameter sets—a theory is stable if $|S_n(A)| \leq |A| + \aleph_0$ for all parameter sets $A$ and all $n$
• No order property meaning stable theories cannot define infinite linear orders; this is the key combinatorial characterization
• Rich structure theory including forking independence, weight, and canonical bases; stable theories are the "tame" theories in Shelah's classification program

$\omega$-Categorical Models

• Unique countable model up to isomorphism—the theory has exactly one countable model, making it maximally determined at the countable level
• Equivalent to finite automorphism orbits by the Ryll-Nardzewski theorem; every $n$-type is isolated
• Examples include dense linear orders without endpoints, the random graph, and vector spaces over finite fields

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Quick Reference Table

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Self-Check Questions

1. What property do both saturated and atomic models share in terms of their relationship to types, and how do they differ in what types they realize?

2. If a theory has a prime model and a countable saturated model, what can you conclude about whether these models are isomorphic? Under what conditions would they coincide?

3. Which two model types are characterized by embedding relationships, and how do their embedding directions differ?

4. Compare and contrast strongly minimal theories and stable theories: what structural constraint defines each, and why is every strongly minimal theory stable but not vice versa?

5. An FRQ asks you to identify a model type that combines "richness" with "symmetry." Which model type satisfies both properties, and what theorem guarantees this connection for models of complete theories?