Journal of Symbolic Logic

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

Assaf Rinot

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

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.

Article information

Source
J. Symbolic Logic Volume 75, Issue 3 (2010), 1035-1065.

Dates
First available in Project Euclid: 9 July 2010

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1278682214

Digital Object Identifier
doi:10.2178/jsl/1278682214

Mathematical Reviews number (MathSciNet)
MR2723781

Zentralblatt MATH identifier
1203.03074

Subjects
Primary: 03E35: Consistency and independence results
Secondary: 03E05: Other combinatorial set theory

Keywords
diamond diamond star saturation approachability ideal weak square reflection principles stationary hitting sap

Citation

Rinot, Assaf. A relative of the approachability ideal, diamond and non-saturation. J. Symbolic Logic 75 (2010), no. 3, 1035--1065. doi:10.2178/jsl/1278682214. https://projecteuclid.org/euclid.jsl/1278682214


Export citation