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A relative of the approachability ideal, diamond and non-saturation

Published online by Cambridge University Press:  12 March 2014

Assaf Rinot
Assaf Rinot*
Affiliation:
School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978. Israel. E-mail: assaf@rinot.com, URL: http://www.tau.ac.il/~rinot
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Abstract

Let λ denote a singular cardinal. Zeman, improving a previous result of Shelah, proved that together with 2λ = λ+ implies ⋄S for every S ⊆ λ+ that reflects stationarily often.

In this paper, for a set S ⊆ λ+, a normal subideal of the weak approachability ideal is introduced, and denoted by I[S; λ]. We say that the ideal is fat if it contains a stationary set. It is proved:

1. if I[S; λ] is fat, then NSλ + ∣ S is non-saturated;

2. if I[S; λ] is fat and 2λ = λ+, then ⋄S holds;

3. implies that I[S; λ] is fat for every S ⊆ λ+ that reflects stationarily often;

4. it is relatively consistent with the existence of a supercompact cardinal that fails, while I[S; λ] is fat for every stationary S ⊆ λ+ that reflects stationarily often.

The stronger principle is studied as well.

Keywords

diamonddiamond starsaturationapproachability idealweak squarereflection principlesstationary hittingsap
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 75 , Issue 3 , September 2010 , pp. 1035 - 1065
Copyright
Copyright © Association for Symbolic Logic 2010

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