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. (shrink)