### Abstract

Starting with a λ-supercompact cardinal κ, where λ is a regular cardinal greater than or equal to κ, we produce a model with a stationary subset S of P_{κ} λ such that NS_{κ λ} | S, the ideal generated by the non-stationary ideal NS_{κ λ} over P_{κ} λ together with P_{κ} λ {set minus} S, is λ^{+}-saturated. Using this model we prove the consistency of the existence of such a stationary set together with the Generalized Continuum Hypothesis (GCH). We also show that in our model we can make NS_{κ λ} | S (κ, λ) λ^{+}-saturated, where S (κ, λ) is the set of all x ∈ P_{κ} λ such that ot (x), the order type of x, is a regular cardinal and x is stationary in sup (x). Furthermore we construct a model where NS_{κ λ} | S (κ, λ) is κ^{+}-saturated but GCH fails. We show that if S {set minus} S (κ, λ) is stationary in P_{κ} λ, then S can be split into λ many disjoint stationary subsets.

Original language | English |
---|---|

Pages (from-to) | 100-123 |

Number of pages | 24 |

Journal | Annals of Pure and Applied Logic |

Volume | 149 |

Issue number | 1-3 |

DOIs | |

Publication status | Published - 2007 Nov |

Externally published | Yes |

### Fingerprint

### Keywords

- Club-shooting
- GCH
- Non-stationary ideal
- P λ
- Saturated ideal

### ASJC Scopus subject areas

- Logic

### Cite this

**Local saturation of the non-stationary ideal over P _{κ} λ.** / Usuba, Toshimichi.

Research output: Contribution to journal › Article

_{κ}λ',

*Annals of Pure and Applied Logic*, vol. 149, no. 1-3, pp. 100-123. https://doi.org/10.1016/j.apal.2007.08.002

}

TY - JOUR

T1 - Local saturation of the non-stationary ideal over Pκ λ

AU - Usuba, Toshimichi

PY - 2007/11

Y1 - 2007/11

N2 - Starting with a λ-supercompact cardinal κ, where λ is a regular cardinal greater than or equal to κ, we produce a model with a stationary subset S of Pκ λ such that NSκ λ | S, the ideal generated by the non-stationary ideal NSκ λ over Pκ λ together with Pκ λ {set minus} S, is λ+-saturated. Using this model we prove the consistency of the existence of such a stationary set together with the Generalized Continuum Hypothesis (GCH). We also show that in our model we can make NSκ λ | S (κ, λ) λ+-saturated, where S (κ, λ) is the set of all x ∈ Pκ λ such that ot (x), the order type of x, is a regular cardinal and x is stationary in sup (x). Furthermore we construct a model where NSκ λ | S (κ, λ) is κ+-saturated but GCH fails. We show that if S {set minus} S (κ, λ) is stationary in Pκ λ, then S can be split into λ many disjoint stationary subsets.

AB - Starting with a λ-supercompact cardinal κ, where λ is a regular cardinal greater than or equal to κ, we produce a model with a stationary subset S of Pκ λ such that NSκ λ | S, the ideal generated by the non-stationary ideal NSκ λ over Pκ λ together with Pκ λ {set minus} S, is λ+-saturated. Using this model we prove the consistency of the existence of such a stationary set together with the Generalized Continuum Hypothesis (GCH). We also show that in our model we can make NSκ λ | S (κ, λ) λ+-saturated, where S (κ, λ) is the set of all x ∈ Pκ λ such that ot (x), the order type of x, is a regular cardinal and x is stationary in sup (x). Furthermore we construct a model where NSκ λ | S (κ, λ) is κ+-saturated but GCH fails. We show that if S {set minus} S (κ, λ) is stationary in Pκ λ, then S can be split into λ many disjoint stationary subsets.

KW - Club-shooting

KW - GCH

KW - Non-stationary ideal

KW - P λ

KW - Saturated ideal

UR - http://www.scopus.com/inward/record.url?scp=35348992040&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=35348992040&partnerID=8YFLogxK

U2 - 10.1016/j.apal.2007.08.002

DO - 10.1016/j.apal.2007.08.002

M3 - Article

VL - 149

SP - 100

EP - 123

JO - Annals of Pure and Applied Logic

JF - Annals of Pure and Applied Logic

SN - 0168-0072

IS - 1-3

ER -