Two Boolean algebras are elementarily equivalent if and only if they satisfy the same first-order statements in the language of Boolean algebras. We prove that every Boolean algebra is elementarily equivalent to the algebra of clopen subsets of a normal P-space.

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 ${\omega }_1$ 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 $2^{\omega }$ nonisomorphic, countable models. Thus we answer this conjecture in the negative for any complete...