On Shattering, Splitting and Reaping Partitions

✍ Scribed by Lorenz Halbeisen

• Publisher: John Wiley and Sons
• Year: 1998
• Tongue: English
• Weight: 662 KB
• Volume: 44
• Category: Article
• ISSN: 0044-3050
• DOI: 10.1002/malq.19980440109

✦ Synopsis

Abstract

In this article we investigate the dual‐shattering cardinal ℌ, the dual‐splitting cardinal 𝔖 and the dual‐reaping cardinal 𝔎, which are dualizations of the well‐known cardinals 𝔥 (the shattering cardinal, also known as the distributivity number of P(ω)/fin), s (the splitting number) and 𝔠 (the reaping number). Using some properties of the ideal 𝔍 of nowhere dual‐Ramsey sets, which is an ideal over the set of partitions of ω, we show that add(𝔍) = cov(𝔍) = 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 t, where t is the tower number. Concerning 𝔖 we show that cov(M) ⩽ 𝔖 𝔯 (where M is the ideal of the meager sets). For the dual‐reaping cardinal 𝔖 we get p 𝔖 ⩽ 𝔭 ⩽ 𝔯 (where 𝔭 is the pseudo‐intersection number) and for a modified dual‐reaping number 𝔖′ we get 𝔖′ ⩽ 𝔬 (where 𝔬 is the dominating number). As a consistency result we get 𝔖 < cov(𝔐).