Decidability of the AE-theory of the lattice of $${\varPi }_1^0$$ Π 1 0 classes

Abstract

An AE-sentence is a sentence in prenex normal form with all universal quantifiers preceding all existential quantifiers, and the AE-theory of a structure is the set of all AE-sentences true in the structure. We show that the AE-theory of \((\mathscr {L}({\varPi }_1^0), \cap , \cup , 0, 1)\) is decidable by giving a procedure which, for any AE-sentence in the language, determines the truth or falsity of the sentence in our structure.