Einstein Completeness as Categoricity

Lecture 4: Categoricity implies Completeness

In first order logic, the theory of a structure is a well-defined object; here such a theory is not so clearly specified. An infinite conjunction of first order sentences behaves very much like a single sentence; in particular it satisfies both the upward and downward Löwenheim Skolem theorems. In contrast, the conjunction of all Lω1,ω true in an uncountable model may not have a countable model...

Completeness and Categoricity (in power): Formalization without Foundationalism

We propose a criterion to regard a property of a theory (in first or second order logic) as virtuous: the property must have significant mathematical consequences for the theory (or its models). We then rehearse results of Ajtai, Marek, Magidor, H. Friedman and Solovay to argue that for second order logic, ‘categoricity’ has little virtue. For first order logic, categoricity is trivial; but ‘ca...

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 ...