### Canonical models for $\aleph_1$ combinatorics

by Shelah and Zapletal. [ShZa:610]

Annals Pure and Applied Logic, 1999

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.

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