Some pathological examples of precipitous ideals.

*(English)*Zbl 1140.03032Summary: 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 \(0^{\dagger }\).

\(\bullet\) 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
Full Text:
DOI

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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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