Categoricity of computable infinitary theories

Abstract

Computable structures of Scott rank \({\omega_1^{CK}}\) are an important boundary case for structural complexity. While every countable structure is determined, up to isomorphism, by a sentence of \({\mathcal{L}_{\omega_1 \omega}}\), this sentence may not be computable. We give examples, in several familiar classes of structures, of computable structures with Scott rank \({\omega_1^{CK}}\) whose computable infinitary theories are each \({\aleph_0}\)-categorical. General conditions are given, covering many known methods for constructing computable structures with Scott rank \({\omega_1^{CK}}\), which guarantee that the resulting structure is a model of an \({\aleph_0}\)-categorical computable infinitary theory.