11.2 Morley's categoricity theorem

Morley's categoricity theorem is a game-changer in model theory. It says if a theory in a countable language is categorical in one uncountable cardinality, it's categorical in all of them. This links the number of non-isomorphic models a theory has to their size.

The theorem sparked the development of stability theory, a key area in modern model theory. It introduced important concepts like ω-stability and indiscernibles, which are now fundamental in analyzing theories and their models. This work has far-reaching impacts, influencing fields from algebra to set theory.

Morley's Categoricity Theorem

Theorem Statement and Key Concepts

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• Morley's categoricity theorem states if a theory in a countable language is categorical in some uncountable cardinality, it is categorical in all uncountable cardinalities
• Theorem addresses relationship between number of non-isomorphic models a theory has and cardinality of those models
• Categoricity refers to existence of a unique model (up to isomorphism) of a given cardinality
• Countable cardinalities include finite sets and $\aleph_0$ (natural numbers)
• Uncountable cardinalities start from $\aleph_1$ (real numbers) and continue upwards
• Isomorphism between models preserves structure and relationships between elements

Proof Techniques and Components

• Proof involves complex techniques from model theory (saturated models, indiscernibles)
• Stability plays crucial role in proof, making Morley's theorem cornerstone of stability theory
• Typical proof structure:
  • Show theory categorical in one uncountable cardinality must be ω-stable
  • Use ω-stability to prove categoricity in all uncountable cardinalities
• Saturated models used to construct isomorphisms between models of different cardinalities
• Indiscernibles employed to establish structural properties of models
• ω-stability implies existence of prime models over arbitrary sets

Limitations and Generalizations

• Theorem applies only to theories in countable languages
• Does not address categoricity in countable cardinalities
• Shelah's work extended results to categoricity in successor cardinals
• Zilber's work connected theorem to strongly minimal sets in algebraic geometry
• Baldwin-Lachlan theorem characterizes uncountably categorical theories using prime and saturated models
• Morley rank and Morley degree developed as tools for analyzing ω-stable theories

Applying Morley's Theorem

Preliminary Analysis

• Thoroughly understand theory in question (language, axioms, models)
• Identify if language of theory is countable (finite or countably infinite symbols)
• Determine if theory has infinite models (necessary for uncountable categoricity)
• Analyze structure of models to identify potential isomorphisms or differences
• Consider automorphism group of models for insights into categoricity

Proving Categoricity

• Determine if theory is categorical in at least one uncountable cardinality
• Prove all models of that cardinality are isomorphic using techniques:
  • Back-and-forth arguments (construct isomorphism by extending partial maps)
  • Analysis of definable sets (show definable sets have same structure across models)
  • Examination of automorphism groups (use transitivity properties)
• Establish uniqueness of models up to isomorphism for chosen cardinality
• If categoricity proven for one uncountable cardinality, invoke Morley's theorem

Challenges and Considerations

• Theories with uncountable languages not covered by Morley's theorem
• Theories categorical only in countable cardinalities require different analysis
• Non-elementary classes may exhibit different categoricity behavior
• Consider potential counterexamples or limitations specific to given theory
• Familiarity with known categorical theories provides context:
  • Theory of algebraically closed fields of fixed characteristic (uncountably categorical)
  • Theory of dense linear orders without endpoints (not uncountably categorical)

Morley's Theorem in Stability Theory

Foundational Impact

• Morley's theorem marks beginning of modern model theory
• Introduced concept of ω-stability, central to classification of first-order theories
• ω-stability defined as having countable number of complete types over any countable set of parameters
• Theorem motivated study of various stability notions:
  • Superstability (stable in all uncountable cardinalities)
  • Strict stability (stable but not superstable)
• Proof techniques, particularly use of indiscernibles, influenced subsequent work
• Indiscernibles refer to sets of elements satisfying same formulas under permutation

Development of Stability Theory

• Morley's work led to study of categoricity spectra
• Categoricity spectrum of a theory describes cardinalities where it is categorical
• Shelah extended results to categoricity in successor cardinals
• Stability spectrum theorem developed, relating stability to number of models
• Forking independence introduced as generalization of algebraic independence
• Classification theory emerged, aiming to understand structure of models based on stability-theoretic properties

Connections and Applications

• Theorem's connection to saturated models significant in stability theory
• Saturated models serve as universal domains for embedding other models
• Morley rank developed as measure of complexity for definable sets in ω-stable theories
• Stability theory found applications beyond model theory:
  • Algebraic geometry (strongly minimal sets)
  • Combinatorics (Ramsey theory)
  • Group theory (stable groups)

Morley's Theorem: History and Significance

Historical Context

• Morley's theorem, published in 1965, resolved conjecture posed by Łoś in 1954
• Łoś conjectured relationship between categoricity in different uncountable cardinalities
• State of model theory before Morley's theorem:
  • Focus on general results about logic (completeness, compactness)
  • Work of Tarski on elementary classes and algebraically closed fields
  • Vaught's two-cardinal theorem relating models of different cardinalities

Impact on Model Theory

• Marked shift from general logic to study of specific classes of theories and structures
• Introduced new techniques and concepts:
  • Morley sequences (indiscernible sequences in stable theories)
  • Morley rank (measure of complexity for definable sets)
  • Prime models over sets (minimal elementary extensions)
• Influenced direction of research for decades following publication
• Led to development of stability theory as major branch of model theory
• Inspired classification theory, aiming to understand theories based on model-theoretic properties

Broader Mathematical Influence

• Impact extended beyond model theory to other areas of mathematics
• Connections with algebraic geometry:
  • Zilber's work on strongly minimal sets
  • Geometric stability theory developed by Hrushovski and others
• Influenced study of infinitary logics and abstract elementary classes
• Applications in algebra:
  • Understanding of algebraically closed fields
  • Study of differentially closed fields
• Connections to set theory through study of large cardinal axioms and their model-theoretic consequences