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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
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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

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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