CANONICAL MODELS FOR ℵ_{1}-COMBINATORICS

SAHARON SHELAH JINDRICH ZAPLETAL

Abstract. We define the property of Π_{2}-compactness of a statement ϕ of set theory, meaning roughly that the hard core of the impact of ϕ on combinatorics of ℵ_{1} can be isolated in a canonical model for the statement ϕ. We show that the following statements are Π_{2}-compact: “dominating number= ℵ_{1},” “cofinality of the meager ideal= ℵ_{1}”, “cofinality of the null ideal= ℵ_{1}”, existence of various types of Souslin trees and variations on uniformity of measure and category= ℵ_{1}. The sentence “𝔟 = ℵ_{1}” is discussed as well. Several important new metamathematical patterns among classical statements of set theory are pointed out.