S 5 × S 5 × S 5 lacks the finite model property

It follows from algebraic results of Maddux that every multi-modal logic L such that [S5,S5, . . . ,S5] ⊆ L ⊆ S5 is undecidable, whenever n ≥ 3. This implies that the product logic S5×S5×S5 does not have the product finite model property. Here we answer a question of Gabbay and Shehtman by showing that S5×S5×S5 also lacks the ‘real’ finite model property (fmp). We prove that every logic L from the above interval lacks the fmp. (In algebraic setting: If V is a variety of n-dimensional diagonalfree cylindric algebras which contains all the representables then V does not have the finite algebra property.)