Topology on the set of maximal consistent propositional theories and the Cantor ternary set
WANG Guo-jun1, WANG Wei2, SONG Jian-she3
(1 College of Mathematics and Information Science, Shaanxi Normal University, Xi′an 710062, Shaanxi, China; 2 Center of Information and Educational Technologies, Xi′an University of Finance and Economics, Xi′an 710061, Shaanxi, China;3 Xi′an High-Technical Institute, Xi′an 710025, Shaanxi, China)
Abstract:
Clearly described the structure of the set consisting of all maximal consistent propositional theories, and it is proved that a logic theory is maximal consistent if and only if it is a closure of a sequence of literals. In a natural way, a compact Hausdorff topology is introduced, and it is proved that the obtained topological space is homeomorphic to the Cantor ternary set. As an application, a simple proof of a complete theorem is proposed.
KeyWords:
mathematical logic; maximal consistent theory; compact Hausdorff standard topology; Cantor ternary set; completeness
