Saturated Models

Origin. Morley, Vaught (1950s-60s). Realize all types. Universal homogeneity. Rich structures. Foundation for stability theory.

Models. κ-saturated: realizes all types over sets < κ. Every consistent extension realized. Ultrapowers give saturation. Monster models.

Formalism.

Type: p(x) = consistent set of formulas with parameter x. Complete: decides all formulas.

Realizes: M realizes p if some a ∈ M satisfies all φ ∈ p. p(a) holds.

κ-saturated: M is κ-saturated iff: for all A ⊆ M with |A| < κ, M realizes all types over A.

ω-saturated: Realizes types over finite sets. Countable saturation.

Existence: κ-saturated models exist (with choice). Ultrapower construction. Directed limits.

Homogeneity: κ-saturated → κ-homogeneous. Partial isomorphisms extend.

Monster model: Very saturated and strongly homogeneous. "Universal domain." Contains all small models.

Applications: Definability: if definable over all sets, globally definable. Automorphisms: many in saturated. Saturation + categoricity = uniqueness.

Ultrapower: ∏_U M / ultrafilter. ω₁-saturated (with countable index). Łoś theorem.

Symbols.

Symbol	Unicode	Name	Meaning
M^sat	—	Saturated	Realizes types
p(x)	—	Type	Consistent formulas
κ	U+03BA	Cardinality	Saturation level

Metatheory. Existence requires choice. Uniqueness (up to iso) for saturated of same cardinality. Transfer principles.

Applies to. Stability theory. Classification. Non-standard analysis. Model constructions.

Limitations. Requires choice. Size issues. Existence conditions. Technical complexity.