Scattered and hereditarily irresolvable spaces in modal logic

When we interpret modal ♦ as the limit point operator of a topological space, the Gödel-Löb modal system GL defines the class Scat of scattered spaces. We give a partition of Scat into α-slices Sα , where α ranges over all ordinals. This provides topological completeness and definability results for extensions of GL. In particular, we axiomatize the modal logic of each ordinal α, thus obtaining a simple proof of the Abashidze–Blass theorem. On the other hand, when we interpret ♦ as closure in a topological space, the Grzegorczyk modal system Grz defines the class HI of hereditarily irresolvable spaces. We also give a partition of HI into α-slices Hα , where α ranges over all ordinals. For a subset A of a hereditarily irresolvable space X and an ordinal α, we introduce the α-representation of A, give an axiomatization of the α-representation of A, and characterize Hα in terms of α-representations. We prove that X ∈ H1 iff X is submaximal. For a positive integer n, we generalize the notion of a submaximal space to that of an n-submaximal space, and prove that X ∈ Hn iff X is n-submaximal. This provides topological completeness and definability results for extensions of Grz. We show that the two partitions are related to each other as follows. For a successor ordinal α = β + n, with β a limit ordinal and n a positive integer, we have Hα ∩ Scat = Sβ+2n−1 ∪ Sβ+2n , and for a limit ordinal α, we have Hα ∩ Scat = Sα . As a result, we obtain full and faithful translations of ordinal complete extensions of Grz into ordinal complete extensions of GL, thus generalizing the Kuznetsov–Goldblatt–Boolos theorem. G. Bezhanishvili (B) · P. J. Morandi Department of Mathematical Sciences, New Mexico State University, Las Cruces, NM 88003-8001, USA e-mail: [email protected] P. J. Morandi e-mail: [email protected]