The Tarski–Lindenbaum algebra of the class of strongly constructivizable models with \(\omega \)-stable theories

We study the class of all strongly constructivizable models having \(\omega \)-stable theories in a fixed finite rich signature. It is proved that the Tarski–Lindenbaum algebra of this class considered together with a Gödel numbering of the sentences is a Boolean \(\Sigma ^1_1\)-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 all Boolean \(\Sigma ^1_1\)-algebras. This gives a characterization to the Tarski-Lindenbaum algebra of the class of all strongly constructivizable models with \(\omega \)-stable theories.