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There is a conjecture of Vaught [17] which states: Without The Generalized Continuum Hypothesis one can prove the existence of a complete theory with exactly ω1 nonisomorphic, denumerable models. In this paper we show that there is no such theory in the class of complete extensions of the theory of Boolean algebras. More precisely, any complete extension of the theory of Boolean algebras has either 1 or 2 nonisomorphic, countable models. Thus we answer this conjecture in the negative for any complete extension of the theory of Boolean algebras. In Rosenstein [15] there is a similar conjecture that any countable complete theory which has uncountably many denumerable models must have 2 nonisomorphic denumerable models, and this is true without using the Continuum Hypothesis. This paper is an excerpt of the author’s thesis, which was written under the guidance of Professor G. C. Nelson. A more detailed exposition of the material may be found there.

The number of countable isomorphism types of complete extensions of the theory of Boolean algebras