Notes on countably generated complete Boolean algebras

TL;DR

This paper characterizes when atomless Boolean algebras are countably generated, offers new proofs of known results, and demonstrates how Jensen's coding theorem can produce large cardinality examples, answering a longstanding question.

Contribution

It provides a necessary and sufficient condition for countably generated atomless Boolean algebras and applies Jensen's coding theorem to construct large examples, extending prior results.

Findings

Characterization of countably generated atomless Boolean algebras

New proofs of classical results by Gaifman-Hales, Solovay, Jech, Kunen, and Magidor

Construction of large cardinality Boolean algebras using Jensen's coding theorem

Abstract

We give a necessary and sufficient condition for an atomless Boolean algebra to be countably generated, and use it to give new proofs of some some know facts due to Gaifman-Hales and Solovay and also due to Jech, Kunen and Magidor. We also show that Jensen's coding theorem can be used to provide cardinal preserving countably generated complete Boolean algebras of arbitrary large cardinality. This answers a question of Jech [5] from 1976.