# Sh:610

- Shelah, S., & Zapletal, J. (1999).
*Canonical models for \aleph_1-combinatorics*. Ann. Pure Appl. Logic,**98**(1-3), 217–259. arXiv: math/9806166 DOI: 10.1016/S0168-0072(98)00022-0 MR: 1696852 -
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. - 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}, doi = {10.1016/S0168-0072(98)00022-0}, mrclass = {03E05 (03E17 03E40 03E45 03E50 03E60)}, mrnumber = {1696852}, mrreviewer = {Pierre Matet}, doi = {10.1016/S0168-0072(98)00022-0}, note = {\href{https://arxiv.org/abs/math/9806166}{arXiv: math/9806166}}, arxiv_number = {math/9806166} }