On almost precipitous ideals

Archive for Mathematical Logic 49 (3):301-328 (2010)
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Abstract

With less than 0# two generic extensions ofL are identified: one in which ${\aleph_1}$ , and the other ${\aleph_2}$ , 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 ${f:\omega_1 \to \omega_1}$ with ${\|f\|=\omega_2}$ , then after ${Col(\aleph_2,\delta)}$ there is a normal precipitous ideal over ${\aleph_1}$ . The existence of a pseudo-precipitous ideal over a successor cardinal is shown to give an inner model with a strong cardinal

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Citations of this work

On the strength of no normal precipitous filter.Moti Gitik & Liad Tal - 2011 - Archive for Mathematical Logic 50 (1-2):223-243.
More on the pressing down game.Jakob Kellner & Saharon Shelah - 2011 - Archive for Mathematical Logic 50 (3-4):477-501.
A model with a precipitous ideal, but no normal precipitous ideal.Moti Gitik - 2013 - Journal of Mathematical Logic 13 (1):1250008.

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References found in this work

[Omnibus Review].Thomas Jech - 1992 - Journal of Symbolic Logic 57 (1):261-262.
Collapsing functions.Ernest Schimmerling & Boban Velickovic - 2004 - Mathematical Logic Quarterly 50 (1):3-8.
Proper and Improper Forcing.Péter Komjáath - 2000 - Studia Logica 64 (3):421-425.

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