Universal Boolean Algebras with Applications to Semantic Classes of Models

We explore numbered Boolean algebras over levels \(\varXi \) of arithmetical and analytical hierarchies. We show the existence and uniqueness (up to computable isomorphism) of universal Boolean \(\varXi \) -algebras, determine the levels in which such algebras exist, and classify the universal algebras up to isomorphism. We apply these results to the semantic class of all countable saturated models having decidable \(\omega \) -stable theories in a fixed finite rich signature. It turns out that the Tarski-Lindenbaum algebra of this class equipped with a Gödel numbering of the sentences is a Boolean \(\varSigma ^1_1\) -algebra whose computable ultrafilters form a dense subset in the set of all ultrafilters; moreover, this algebra is universal with respect to the class of Boolean \(\varSigma ^1_1\) -algebras. This determines uniquely the isomorphism type of this Boolean algebra.