Gitik, Moti Some pathological examples of precipitous ideals. (English) Zbl 1140.03032 J. Symb. Log. 73, No. 2, 492-511 (2008). Summary: 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 [“Smoke and mirrors: combinatorial properties of small cardinals equiconsistent with huge cardinals” (to appear)] and R. Laver [Isr. J. Math. 48, 97–108 (1984; Zbl 0595.03044)], respectively. The present examples differ in two ways: first, they use only a measurable cardinal and, second, the 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 \(\kappa \) such that\(\bullet\) after the forcing with its positive sets the cardinality of \(\kappa \) remains above \(\aleph _{1}\)\(\bullet\) there is no normal precipitous ideal then there is \(0^{\dagger }\). Cited in 3 Documents MSC: 03E35 Consistency and independence results 03E05 Other combinatorial set theory 03E55 Large cardinals Keywords:precipitous ideal; measurable cardinal; normal ideal Citations:Zbl 0595.03044 PDF BibTeX XML Cite \textit{M. Gitik}, J. Symb. Log. 73, No. 2, 492--511 (2008; Zbl 1140.03032) Full Text: DOI OpenURL References: [1] DOI: 10.1007/BF02761155 · Zbl 0595.03044 [2] Handbook of set theory [3] On generic elementary embeddings 54 pp 700– (1989) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.