On the elementary equivalence of automorphism groups of Boolean algebras; downward Skolem löwenheim theorems and compactness of related quantifiers

Journal of Symbolic Logic 45 (2):265-283 (1980)
Abstract
THEOREM 1. (⋄ ℵ 1 ) If B is an infinite Boolean algebra (BA), then there is B 1 such that $|\operatorname{Aut} (B_1)| \leq B_1| = \aleph_1$ and $\langle B_1, \operatorname{Aut} (B_1)\rangle \equiv \langle B, \operatorname{Aut}(B)\rangle$ . THEOREM 2. (⋄ ℵ 1 ) There is a countably compact logic stronger than first-order logic even on finite models. This partially answers a question of H. Friedman. These theorems appear in §§ 1 and 2. THEOREM 3. (a) (⋄ ℵ 1 ) If B is an atomic ℵ 1 -saturated infinite BA, ψ ε L ω 1ω and $\langle B, \operatorname{Aut} (B)\rangle \models\psi$ then there is B 1 such that $|\operatorname{Aut}(B_1)| \leq |B_1| = \aleph_1$ and $\langle B_1, \operatorname{Aut}(B_1)\rangle\models\psi$ . In particular if B is 1-homogeneous so is B 1 . (b) (a) holds for B = P(ω) even if we assume only CH
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