## Journal of Symbolic Logic

### On the Elementary Equivalence of Automorphism Groups of Boolean Algebras; Downward Skolem Lowenheim Theorems and Compactness of Related Quantifiers

#### Abstract

THEOREM 1. $(\Diamond_{\aleph_1})$ 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. $(\Diamond_{\aleph_1})$ 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 $\S\S 1$ and 2. THEOREM 3. (a) $(\Diamond_{\aleph_1})$ If B is an atomic $\aleph_1$-saturated infinite $BA, \psi \epsilon L_{\omega 1\omega}$ 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 $B = P(\omega)$ even if we assume only CH.

#### Article information

Source
J. Symbolic Logic, Volume 45, Issue 2 (1980), 265-283.

Dates
First available in Project Euclid: 6 July 2007

https://projecteuclid.org/euclid.jsl/1183740558

Mathematical Reviews number (MathSciNet)
MR569397

Zentralblatt MATH identifier
0445.03012

JSTOR