Canonical models for aleph_1 combinatorics

Details Canonical models for aleph_1 combinatorics Canonical models for aleph_1 combinatorics

1998-06-15

arxiv.org/abs/math/9806166v1

Mathematics

Logic

Scientific paper

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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