# Sh:610

• Shelah, S., & Zapletal, J. (1999). Canonical models for \aleph_1-combinatorics. Ann. Pure Appl. Logic, 98(1-3), 217–259.
• Abstract:
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.
• Version 1998-06-20_10 (38p) published version (43p)
Bib entry
@article{Sh:610,
author = {Shelah, Saharon and Zapletal, Jind{\v{r}}ich},
title = {{Canonical models for $\aleph_1$-combinatorics}},
journal = {Ann. Pure Appl. Logic},
fjournal = {Annals of Pure and Applied Logic},
volume = {98},
number = {1-3},
year = {1999},
pages = {217--259},
issn = {0168-0072},
mrnumber = {1696852},
mrclass = {03E05 (03E17 03E40 03E45 03E50 03E60)},
doi = {10.1016/S0168-0072(98)00022-0},
note = {\href{https://arxiv.org/abs/math/9806166}{arXiv: math/9806166}},
arxiv_number = {math/9806166}
}