An example related to Gregory’s Theorem

Abstract

In this paper, we give an example of a complete computable infinitary theory T with countable models \({\mathcal{M}}\) and \({\mathcal{N}}\) , where \({\mathcal{N}}\) is a proper computable infinitary extension of \({\mathcal{M}}\) and T has no uncountable model. In fact, \({\mathcal{M}}\) and \({\mathcal{N}}\) are (up to isomorphism) the only models of T. Moreover, for all computable ordinals α, the computable \({\Sigma_\alpha}\) part of T is hyperarithmetical. It follows from a theorem of Gregory (JSL 38:460–470, 1972; Not Am Math Soc 17:967–968, 1970) that if T is a Π11 set of computable infinitary sentences and T has a pair of models \({\mathcal{M}}\) and \({\mathcal{N}}\) , where \({\mathcal{N}}\) is a proper computable infinitary extension of \({\mathcal{M}}\) , then T would have an uncountable model.