Hostname: page-component-76fb5796d-2lccl Total loading time: 0 Render date: 2024-04-27T13:38:35.555Z Has data issue: false hasContentIssue false
  • English
  • Français

Some pathological examples of precipitous ideals

Published online by Cambridge University Press:  12 March 2014

Moti Gitik
Moti Gitik*
Affiliation:
School of Mathematical Sciences, Raymond and Beverly Sackler Faculty of Exact Science, Tel Aviv University, Ramat Aviv 69978, Israel
Get access Rights & Permissions [Opens in a new window]

Abstract

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 [2] and R. Laver [4] 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 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†.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 73 , Issue 2 , June 2008 , pp. 492 - 511
Copyright
Copyright © Association for Symbolic Logic 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Foreman, M., Ideals and generic elementary embeddings, Handbook of set theory, to appear.CrossRefGoogle Scholar
[2]Foreman, M., Smoke and mirrors: combinatorial properties of small cardinals equiconsistent with huge cardinals, to appear.Google Scholar
[3]Gitik, M., On generic elementary embeddings, this Journal, vol. 54 (1989), no. 3, pp. 700–707.Google Scholar
[4]Laver, R., Precipitousness in forcing extensions, Israel Journal of Mathematics, vol. 48 (1984), no. 2-3, pp. 97–108.CrossRefGoogle Scholar
[5]Levinski, J.-P., These du Troisieme Cycle, Universite Paris VII, Paris, 1980.Google Scholar
[6]Mitchell, W., in Handbook of Set Theory, to appear.Google Scholar