科研之友
微信
新浪微博
Facebook
分享链接
摘要
When we interpret modal lozenge as the limit point operator of a topological space, the Godel-Lob modal system GL defines the class Scat of scattered spaces. We give a partition of Scat into alpha-slices S(alpha), where alpha 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 alpha, thus obtaining a simple proof of the Abashidze-Blass theorem. On the other hand, when we interpret lozenge 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 alpha-slices H(alpha), where alpha ranges over all ordinals. For a subset A of a hereditarily irresolvable space X and an ordinal alpha, we introduce the alpha-representation of A, give an axiomatization of the alpha-representation of A, and characterize H(alpha) in terms of alpha-representations. We prove that X is an element of 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 is an element of 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 alpha = beta + n, with beta a limit ordinal and n a positive integer, we have H(alpha) boolean AND Scat = S(beta+2n-1) boolean OR S(beta+2n), and for a limit ordinal alpha, we have H(alpha) boolean AND Scat = 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.
- 出版日期2010-4
