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Some pathological examples of precipitous ideals. (English) Zbl 1140.03032

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 \(0^{\dagger }\).

MSC:

03E35 Consistency and independence results
03E05 Other combinatorial set theory
03E55 Large cardinals

Citations:

Zbl 0595.03044
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References:

[1] DOI: 10.1007/BF02761155 · Zbl 0595.03044
[2] Handbook of set theory
[3] On generic elementary embeddings 54 pp 700– (1989)
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