## Abstract

With less than 0^{#} two generic extensions of L are identified: one in which א_{1}, and the other א_{2}, is almost precipitous. This improves the consistency strength upper bound of almost precipitousness obtained in Gitik M, Magidor M (On partialy wellfounded generic ultrapowers, in Pillars of Computer Science, 2010), and answers some questions raised there. Also, main results of Gitik (On normal precipitous ideals, 2010), are generalized-assumptions on precipitousness are replaced by those on ∞-semi precipitousness. As an application it is shown that if δ is a Woodin cardinal and there is an f: ω_{1} → ω_{2} with {double pipe}f{double pipe} = ω_{2}, then after Col(א_{2}, δ) there is a normal precipitous ideal over א_{1}. The existence of a pseudo-precipitous ideal over a successor cardinal is shown to give an inner model with a strong cardinal.

Original language | English |
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Pages (from-to) | 301-328 |

Number of pages | 28 |

Journal | Archive for Mathematical Logic |

Volume | 49 |

Issue number | 3 |

DOIs | |

State | Published - 2010 |

## Keywords

- Almost precipitous ideals
- Generic ultrapowers
- Well foundedness