Saharon Shelah and Jindrich Zapletal
\def\bb{\frak b} We define the property of $\Pi_2$-compactness of a statement $\phi$ of set theory, meaning roughly that the hard core of the impact of $\phi$ on combinatorics of $\aleph_1$ can be isolated in a canonical model for the statement $\phi$. We show that the following statements are $\Pi_2$-compact: ``dominating number$=\aleph_1$,'' ``cofinality of the meager ideal$=\aleph_1$'', ``cofinality of the null ideal$=\aleph_1$'', existence of various types of Souslin trees and variations on uniformity of measure and category$=\aleph_1$. The sentence ``$\bb=\aleph_1$'' is discussed as well. Several important new metamathematical patterns among classical statements of set theory are pointed out.