Some pathological examples of precipitous ideals

We construct a model with an indecisive precipitous ideal and a model with a precipitous ideal with a non precipitous normal ideal below it. Such kind of examples were previously given by M. Foreman [1] and R. Laver [4] respectively. The present examples differ in two ways: firstthey use only a measurable cardinal and secondthe ideals are over a cardinal. Also a precipitous ideal without a normal ideal below it is constructed. It is shown in addition that if there is a precipitous ideal over a cardinal κ such that • after the forcing with its positive sets the cardinality of κ remains above א1 • there is no a normal precipitous ideal then there is 0†. 1 Indecisive precipitous ideals. In [1] M. Foreman isolated the following natural and wide class of ideals: Definition 1.1 (M. Foreman [1]) Let Z ⊂ P (X) and J be an ideal on Z. Let X ′ ⊂ X and I be the projection of J to an ideal on P (X ′). Then J decides I iff there is a set A ∈ Ĭ and a well ordering W of A and sets A′,W ′, O′, I ′ ∈ V such that for all generic G ⊂ P (Z)/J : 1. An initial segment of the ordinals of V /G is well founded and isomorphic to (|A′|+)V and ∗The main part of this work was done during my visit to UCI. I would like to thank M. Foreman and M. Zeman for their hospitality and to M. Foreman for re-waking my interest in this subject. I am grateful to Assaf Sharon for reading the previous version and asking questions that lead to the present one. Finally I am very grateful to the referee of the paper for his long list of suggestions, corrections and questions.