Decidability of cylindric set algebras of dimension two and first-order logic with two variables

The aim of this paper is to give a new proof for the decidability and finite model property of first-order logic with two variables (without function symbols), using a combinatorial theorem due to Herwig. The results are proved in the framework of polyadic equality set algebras of dimension two (Pse2). The new proof also shows the known results that the universal theory of Pse2 is decidable and that every finite Pse2 can be represented on a finite base. Since the class Cs2 of cylindric set algebras of dimension 2 forms a reduct of Pse2, these results extend to Cs2 as well

Decidability of Cylindric Set Algebras of Dimension Two and First-Order Logic With Two Variables

The aim of this paper is to give a new proof for the decidability and finite model property of first-order logic with two variables (without function symbols), using a combinatorial theorem due to Herwig. The results are proved in the framework of polyadic equality set algebras of dimension two (Pse 2 ). The new proof also shows the known results that the universal theory of Pse 2 is decidable and that every finite Pse 2 can be represented on a finite base. Since the class Cs 2 of cylindric set algebras of dimension 2 forms a reduct of Pse 2 , these results extend to Cs 2 as well. We hasten to remark that the results proved here are not new, and indeed there are several rather different proofs available (references below). We felt justified publishing this new proof, since we believe it is simpler than the proofs known, and accessible to both algebraists and logicians. The proof uses only very elementary ideas from universal algebra and model theory and one heavy combinatorial theor..