Morley’s number of countable models

Abstract

A theory formulated in a countable predicate calculus can have at most 2א0 nonisomorphic countable models. In 1961 R. L. Vaught [9] conjected that if such a theory has uncountably many countable models, then it has exactly 2א0 countable models. This would of course follow immediately if one assumed the continuum hypothesis to be true. Almost ten years later, M. Morley [5] proved that if a countable theory has strictly more than א1 countable models, then it has 2א0 countable models. This leaves us with the possibility that a theory has exactly א1 , but not 2א0 countable models — and even today, Vaught’s question remains unanswered. This paper is an attempt to shed a little light on Morley’s proof.