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On Shattering, Splitting and Reaping Partitions
Author(s) -
Halbeisen Lorenz
Publication year - 1998
Publication title -
mathematical logic quarterly
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.473
H-Index - 28
eISSN - 1521-3870
pISSN - 0942-5616
DOI - 10.1002/malq.19980440109
Subject(s) - mathematics , distributivity , dual (grammatical number) , ideal (ethics) , combinatorics , consistency (knowledge bases) , intersection (aeronautics) , corollary , discrete mathematics , pure mathematics , distributive property , epistemology , philosophy , linguistics , engineering , aerospace engineering
In this article we investigate the dual‐shattering cardinal ℌ, the dual‐splitting cardinal and the dual‐reaping cardinal , which are dualizations of the well‐known cardinals (the shattering cardinal, also known as the distributivity number of P (ω)/fin), s (the splitting number) and (the reaping number). Using some properties of the ideal of nowhere dual‐Ramsey sets, which is an ideal over the set of partitions of ω, we show that add() = cov() = ℌ. With this result we can show that ℌ > ω 1 is consistent with ZFC and as a corollary we get the relative consistency of ℌ > t , where t is the tower number. Concerning we show that cov( M ) ⩽ (where M is the ideal of the meager sets). For the dual‐reaping cardinal we get p ⩽ ⩽ (where is the pseudo‐intersection number) and for a modified dual‐reaping number ′ we get ′ ⩽ (where is the dominating number). As a consistency result we get < cov().