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September 2010 A relative of the approachability ideal, diamond and non-saturation
Assaf Rinot
J. Symbolic Logic 75(3): 1035-1065 (September 2010). DOI: 10.2178/jsl/1278682214

Abstract

Let λ denote a singular cardinal. Zeman, improving a previous result of Shelah, proved that □*λ 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. □*λ 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 □*λ fails, while I[S;λ] is fat for every stationary S⊆λ⁺ that reflects stationarily often.

The stronger principle ♢*λ⁺ is studied as well.

Citation

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Assaf Rinot. "A relative of the approachability ideal, diamond and non-saturation." J. Symbolic Logic 75 (3) 1035 - 1065, September 2010. https://doi.org/10.2178/jsl/1278682214

Information

Published: September 2010
First available in Project Euclid: 9 July 2010

zbMATH: 1203.03074
MathSciNet: MR2723781
Digital Object Identifier: 10.2178/jsl/1278682214

Subjects:
Primary: 03E35
Secondary: 03E05

Rights: Copyright © 2010 Association for Symbolic Logic

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Vol.75 • No. 3 • September 2010
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