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ℙmax variations related to slaloms

Teruyuki Yorioka

Mathematical Logic Quarterly 52 (2):203-216 (2006)

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Canonical models for ℵ1-combinatorics.Saharon Shelah & Jindr̆ich Zapletal - 1999 - Annals of Pure and Applied Logic 98 (1-3):217-259.details We define the property of Π2-compactness of a statement Φ of set theory, meaning roughly that the hard core of the impact of Φ on combinatorics of 1 can be isolated in a canonical model for the statement Φ. We show that the following statements are Π2-compact: “dominating NUMBER = 1,” “cofinality of the meager IDEAL = 1”, “cofinality of the null IDEAL = 1”, “bounding NUMBER = 1”, existence of various types of Souslin trees and variations on uniformity of (...) Download Export citation Bookmark 12 citations
Saturation, Suslin trees and meager sets.Paul Larson - 2005 - Archive for Mathematical Logic 44 (5):581-595.details We show, using a variation of Woodin’s partial order ℙ max , that it is possible to destroy the saturation of the nonstationary ideal on ω 1 by forcing with a Suslin tree. On the other hand, Suslin trees typcially preserve saturation in extensions by ℙ max variations where one does not try to arrange it otherwise. In the last section, we show that it is possible to have a nonmeager set of reals of size ℵ1, saturation of the nonstationary (...) Download Export citation Bookmark 2 citations
The cofinality of the strong measure zero ideal.Teruyuki Yorioka - 2002 - Journal of Symbolic Logic 67 (4):1373-1384.details We give a characterization of the cofinality of the strong measure zero ideal under the continuum hypothesis and prove that we can force it to a value less than the power of the continuum. Download Export citation Bookmark 11 citations
P-points in Qmax models.Q. Feng & W. H. Woodin - 2003 - Annals of Pure and Applied Logic 119 (1-3):121-190.details We show how to get canonical models from in which the nonstationary ideal on ω1 is ω1 dense and there is no P-point. Download Export citation Bookmark 1 citation
An variation for one souslin tree.Paul Larson - 1999 - Journal of Symbolic Logic 64 (1):81-98.details We present a variation of the forcing S max as presented in Woodin [4]. Our forcing is a P max -style construction where each model condition selects one Souslin tree. In the extension there is a Souslin tree T G which is the direct limit of the selected Souslin trees in the models of the generic. In some sense, the generic extension is a maximal model of "there exists a minimal Souslin tree," with T G being this minimal tree. In (...) Download Export citation Bookmark 10 citations
P-points In Models.Q. Feng & W. H. Woodin - 2003 - Annals of Pure and Applied Logic 119 (1-3):121-190.details We show how to get canonical models from in which the nonstationary ideal on ω1 is ω1 dense and there is no P-point. Download Export citation Bookmark 1 citation
< i> P_-points in Qmax models.Qi Feng & W. Hugh Woodin - 2003 - Annals of Pure and Applied Logic 119 (1-3):121-190.details Download Export citation Bookmark 1 citation

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