## Abstract

Let λ denote a singular cardinal. Zeman, improving a previous result of Shelah, proved that □^{+} _{λ} together with 2^{λ} = 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 2^{λ} = 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 language | English |
---|---|

Pages (from-to) | 1035-1065 |

Number of pages | 31 |

Journal | Journal of Symbolic Logic |

Volume | 75 |

Issue number | 3 |

DOIs | |

State | Published - Sep 2010 |

Externally published | Yes |

## Keywords

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