  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. Several important
new metamathematical patterns among classical statements of set theory are
pointed out.
