Scattered and hereditarily irresolvable spaces in modal logic

Abstract

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 S _α , 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 H _α , 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 H _α in terms of α-representations. We prove that ${X \in {\bf H}_{1}}$ 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 ${X \in {\bf H}_{n}}$ 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 ${{\bf H}_{\alpha} \cap {\bf Scat} = {\bf S}_{\beta+2n-1} \cup {\bf S}_{\beta+2n}}$ , and for a limit ordinal α, we have ${{\bf H}_{\alpha} \cap {\bf Scat} = {\bf S}_{\alpha}}$ . 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.