Iterated expansions of models for countable theories and their applications

Abstract

An expansion of a countable model by relations for incomplete types realized in the model is constructed. Theories of models obtained by iterating the expansion defined over countable ordinals are investigated. Theorems concerning the atomicity of countable models in a suitable α-expansion are proved, and we settle the question of whether or not α-expansions have atomic models. A theorem on the realization and omission of generalized types is presented. The resulis obtained are then used to give a direct proof of a theorem of Morley on the number of countable models and to state that Ehrenfeucht theories have finite type rank.