11.1 Categoricity in power and its implications

Categoricity in power is a key concept in model theory, focusing on theories where all models of a specific cardinality are isomorphic. It's closely tied to stability and provides insights into model structure and behavior.

This concept connects to the broader theme of categoricity and completeness by showing how categoricity in uncountable cardinals implies completeness. The Łoś-Vaught Test and Morley's Theorem are crucial in understanding these relationships.

Categoricity in Power

Definition and Significance

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• Categoricity in power refers to a property of a first-order theory where all models of a specific cardinality are isomorphic to each other
• A theory T achieves κ-categoricity when all models of T of cardinality κ are isomorphic
• Stronger condition than mere categoricity, which only requires uniqueness up to isomorphism of models of a specific cardinality
• Characterizes theories with a high degree of uniformity in their models of a given size
• Closely related to the notion of stability in model theory, providing insights into the structure and behavior of models
• Łoś-Vaught Test states that a theory with no finite models that is categorical in some infinite cardinal is complete, highlighting the importance of categoricity in power for completeness results

Examples and Applications

• Natural numbers with addition and multiplication ($(\mathbb{N}, +, \times)$) categorical in power $\aleph_0$ (countable infinity)
• Dense linear orders without endpoints categorical in all uncountable powers (real numbers, rational numbers)
• Complex numbers as an algebraically closed field of characteristic 0 categorical in all uncountable powers
• Infinite-dimensional vector spaces over a fixed field categorical in all uncountable powers

Categoricity and Completeness

Proof of Completeness

• Completeness of a theory means that for any sentence in the language of the theory, either the sentence or its negation is provable from the theory's axioms
• Proof utilizes the Łoś-Vaught Test, connecting categoricity in an uncountable cardinal to completeness
• Key steps involve showing that any two models of the theory are elementarily equivalent using isomorphic elementary submodels of the same uncountable cardinality
• Relies on the downward Löwenheim-Skolem theorem to obtain elementary submodels of the desired cardinality
• Employs transfinite induction to construct an isomorphism between elementary submodels
• Completeness follows from the elementary equivalence of any two models, implying they satisfy exactly the same sentences

Implications and Examples

• Theory of algebraically closed fields of a fixed characteristic is complete and categorical in all uncountable powers
• Theory of dense linear orders without endpoints is complete and categorical in all uncountable powers
• Real closed fields form a complete theory, categorical in all uncountable powers
• Morley's Categoricity Theorem extends completeness to all uncountable cardinals for theories categorical in some uncountable cardinal

Categoricity vs Models

Relationship Analysis

• Categoricity in a given cardinal κ implies exactly one model (up to isomorphism) of cardinality κ
• Morley's Categoricity Theorem states that a theory categorical in some uncountable cardinal is categorical in all uncountable cardinals
• Number of nonisomorphic countable models of a theory categorical in an uncountable cardinal is either 1 or ω (countably infinite)
• Vaught's Conjecture (unproven) posits that a theory in a countable language categorical in one uncountable cardinal is either ω-stable or ω1-categorical
• Spectrum problem investigates possible functions from cardinals to the number of nonisomorphic models of that cardinality
• Shelah's Main Gap Theorem provides a dichotomy for the number of nonisomorphic models in uncountable cardinals, related to stability-theoretic properties of the theory

Examples and Counterexamples

• Theory of dense linear orders without endpoints has exactly two countable models up to isomorphism (rational numbers, real numbers)
• Theory of algebraically closed fields of characteristic 0 has exactly one model of each uncountable cardinality (complex numbers of different sizes)
• Theory of random graphs has 2^κ nonisomorphic models of cardinality κ for every infinite κ
• Theory of equality has exactly one model of each finite cardinality and uncountably many countable models

Categoricity and Automorphisms

Automorphism Group Properties

• Automorphism group of a model consists of all isomorphisms from the model to itself, preserving structure and relations
• For theories categorical in power, the automorphism group often exhibits special properties (transitivity, homogeneity)
• Categoricity in power typically implies the automorphism group acts transitively on tuples of the same type, known as homogeneity
• Ryll-Nardzewski Theorem connects ω-categoricity with oligomorphicity of the automorphism group of the countable model
• For uncountably categorical theories, the automorphism group of the monster model plays a crucial role in studying types and definable sets
• Size and structure of the automorphism group provide insights into symmetry and regularity in models of a categorical theory

Examples and Applications

• Automorphism group of the rational numbers as a dense linear order is highly transitive (can map any n-tuple to any other n-tuple with the same order type)
• Automorphism group of an algebraically closed field of characteristic 0 is simple and acts transitively on algebraically independent sets of the same size
• Automorphism group of the random graph is extremely rich, allowing any finite partial isomorphism to be extended to a full automorphism
• Categoricity often leads to a rich interplay between model-theoretic properties and group-theoretic properties of the automorphism group (oligomorphic groups, permutation groups)