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

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Abstract

Let λ denote a singular cardinal. Zeman, improving a previous result of Shelah, proved that □+ λ together with 2λ = λ+ implies Os 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+f 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 □z.ast; 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.

Original languageEnglish
Pages (from-to)1035-1065
Number of pages31
JournalJournal of Symbolic Logic
Volume75
Issue number3
DOIs
StatePublished - Sep 2010
Externally publishedYes

Keywords

  • Approachability ideal
  • Diamond
  • Diamond star
  • Reflection principles
  • Sap
  • Saturation
  • Stationary hitting
  • Weak square

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