We study computably enumerable boolean algebras, focusing on Stone duality and universality phenomena. We show how classical Stone duality specializes to c.e. boolean algebras, giving a natural bijection between c.e. boolean algebras and $\Pi^0_1$ classes. We also give a new characterization of computably universal-homogeneous c.e. boolean algebras, which yields a more direct proof of the computable isomorphism between the Lindenbaum algebras of theories which satisfy the hypotheses of the second incompleteness theorem.
Stone duality; universal-homogeneous; Logic; boolean algebra; computably enumerable