Boolean powers, recursive models, and the Horn theory of a structure

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 ω 1 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 ω 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 ω 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.1. Preliminaries. Let L be a linear ordering. We define a derivative operation on L in two steps. First we identify any two elements which have a dense linear ordering between them, denoting the resulting ordering by L a . Then in L a we identify any two elements that have only a finite number of elements between them, denoting the result by L 1 . This process can be iterated into the transfinite by taking an appropriate limit construction at limit ordinals. As an example, consider L = (1 + η + 1) · ω + ω 2 · (1 + η + 1) + 1 , L a ∼ = ω + ω 2 · (1 + η + 1) + 1 , L 1 ∼ = 1 + ω · (1 + η + 1) + 1 ∼ = ω · (1 + η + 1) + 1 ,

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Boolean powers, recursive models, and the Horn theory of a structure

Cited by 2 publications

References 24 publications

Model theoretic properties in the variety generated by a primal algebra

Model theoretic properties in the variety generated by a primal algebra

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

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