Tracing Internal Categoricity

Computable Categoricity versus Relative Computable Categoricity

We study the notion of computable categoricity of computable structures, comparing it especially to the notion of relative computable categoricity and its relativizations. We show that every 1-decidable computably categorical structure is relatively ∆2-categorical. We study the complexity of various index sets associated with computable categoricity and relative computable categoricity. We also...

Categoricity of computable infinitary theories

Categoricity and Scott Families

Effective model theory is an area of logic that analyzes the effective content of the typical notions and results of model theory and universal algebra. Typical notions in model theory and universal algebra are languages and structures, theories and models, models and their submodels, automorphisms and isomorphisms, embeddings and elementary embeddings. In this paper languages, structures, and ...

Categoricity and Infinitary Logics

We point out a gap in Shelah’s proof of the following result: Claim 0.1. Let K be an abstract elementary class categorical in unboundedly many cardinals. Then there exists a cardinal λ such that whenever M,N ∈ K have size at least λ, M ≤ N if and only if M L∞,LS(K)+ N . The importance of the claim lies in the following theorem, implicit in Shelah’s work: Theorem 0.2. Assume the claim. Let K be ...

Morley’s Categoricity Theorem

A theory is called κ-categorical, or categorical in power κ, if it has one model up to isomorphism of cardinality κ. Morley’s Categoricity Theorem states that if a theory of first order logic is categorical in some uncountable power κ, then it is categorical in every uncountable power. We provide an elementary exposition of this theorem, by showing that a theory is categorical in some uncountab...