Canonical Models for א 1 - Combinatorics 1

We define the property of Π2-compactness of a statement φ of set theory, meaning roughly that the hard core of the impact of φ on combinatorics of א1 can be isolated in a canonical model for the statement φ. We show that the following statements are Π2compact: “dominating number= א1,” “cofinality of the meager ideal= א1”, “cofinality of the null ideal= א1”,“bounding number= א1”, existence of various types of Souslin trees and variations on uniformity of measure and category= א1. Several important new metamathematical patterns among classical statements of set theory are pointed out. MR subject classification 03E50, 03E40. Secondary 03E35. Research supported by The Israel Science Foundation administered by The Israel Academy of Sciences and Humanities. Publication number 610. Research partially supported by NSF grant number DMS 9022140 and GA ČR grant number 201/97/0216.