TY - JOUR
T1 - Canonical models for א1-combinatorics
AU - Shelah, Saharon
AU - Zapletal, Jindřich
PY - 1999/6/30
Y1 - 1999/6/30
N2 - 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 א1 can be isolated in a canonical model for the statement φ. We show that the following statements are Π2-compact: "dominating number = א1," "cofinality of the meager ideal = א1", "cofinality of the null ideal = א1", "bounding number = א1", existence of various types of Souslin trees and variations on uniformity of measure and category = א1. Several important new metamathematical patterns among classical statements of set theory are pointed out.
AB - 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 א1 can be isolated in a canonical model for the statement φ. We show that the following statements are Π2-compact: "dominating number = א1," "cofinality of the meager ideal = א1", "cofinality of the null ideal = א1", "bounding number = א1", existence of various types of Souslin trees and variations on uniformity of measure and category = א1. Several important new metamathematical patterns among classical statements of set theory are pointed out.
KW - Determinacy
KW - Forcing
UR - http://www.scopus.com/inward/record.url?scp=0033617893&partnerID=8YFLogxK
U2 - 10.1016/S0168-0072(98)00022-0
DO - 10.1016/S0168-0072(98)00022-0
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AN - SCOPUS:0033617893
SN - 0168-0072
VL - 98
SP - 217
EP - 259
JO - Annals of Pure and Applied Logic
JF - Annals of Pure and Applied Logic
IS - 1-3
ER -