Mathematical Logic Quarterly 50 (1):3-8 (2004)

Abstract
We define what it means for a function on ω1 to be a collapsing function for λ and show that if there exists a collapsing function for +, then there is no precipitous ideal on ω1. We show that a collapsing function for ω2 can be added by forcing. We define what it means to be a weakly ω1-Erdös cardinal and show that in L[E], there is a collapsing function for λ iff λ is less than the least weakly ω1-Erdös cardinal. As a corollary to our results and a theorem of Neeman, the existence of a Woodin limit of Woodin cardinals does not imply the existence of precipitous ideals on ω1. We also show that the following statements hold in L[E]. The least cardinal λ with the Chang property ↠ is equal to the least ω1-Erdös cardinal. In particular, if j is a generic elementary embedding that arises from non-stationary tower forcing up to a Woodin cardinal, then the minimum possible value of j is the least ω1-Erdös cardinal
Keywords Large cardinal, precipitous ideal, core model
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DOI 10.1002/malq.200310069
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Citations of this work BETA

On Almost Precipitous Ideals.Asaf Ferber & Moti Gitik - 2010 - Archive for Mathematical Logic 49 (3):301-328.
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More on the Pressing Down Game.Jakob Kellner & Saharon Shelah - 2011 - Archive for Mathematical Logic 50 (3-4):477-501.
Set Forcing and Strong Condensation for H.Liuzhen Wu - 2015 - Journal of Symbolic Logic 80 (1):56-84.
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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