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

  • Assaf Rinot (a1)
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 Sλ+ 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.

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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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