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The entire NS ideal on can be precipitous

Published online by Cambridge University Press:  12 March 2014

Noa Goldring
Noa Goldring*
Affiliation:
Department of Mathematics, Occidental College, Los Angeles, California 90041, USA, E-mail: goldring@oxy.edu
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Extract

The main result of this note is showing that if γ and μ are regular uncountable cardinals with γ ≤ μ then the non-stationary ideal (henceforth the NS ideal) on can be precipitous. This strengthens a result of [1] showing, under the same hypotheses, that a restriction of this ideal can be precipitous. See [1, Theorem 29, p. 36]. In fact, we show that even the strongly NS ideal on is precipitous in our model (since the former ideal is a restriction of the latter, the latter's being precipitous is a stronger assertion).

More precisely, by starting with a model of “ZFC + ‘κ is a supercompact cardinal’ + ‘μ < κ is a regular uncountable cardinal’ ”, we generate a model of ZFC where all cardinals below and including μ are not collapsed and where the NS and strongly NS ideals on Pγμ are precipitous, for all regular uncountable γ which are less than or equal to μ.

As far as consistency strength, we can obtain the same result even if κ is only Woodin in the ground model. However, the proof of this result is more complicated than in the case when κ is a supercompact cardinal. Furthermore, there are essentially no new ideas in adapting the proof relative to a supercompact cardinal to that relative to a Woodin cardinal beyond what appears in, e.g., [2]. We therefore give the complete proof relative to the existence of a supercompact cardinal and then briefly sketch the proof relative to the existence of a Woodin cardinal, using [2] as a reference.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 62 , Issue 4 , December 1997 , pp. 1161 - 1172
Copyright
Copyright © Association for Symbolic Logic 1997

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References

REFERENCES

[1]Foreman, M., Magidor, M., and Shelah, S., Martin's maximum, saturated ideals and non-regular ultrafilters, Part I, Annals of Mathematics, vol. 127 (1988), pp. 1–47.CrossRefGoogle Scholar
[2]Goldring, N., Woodin cardinals and presaturated ideals, Annals of Pure and Applied Logic, vol. 55 (1992), pp. 285–303.CrossRefGoogle Scholar
[3]Jech, T., Set theory, Academic Press, New York, 1978.Google Scholar
[4]Kueker, D. W., Countable approximations and Löwenheim-Skolem theorems, Annals of Mathematical Logic, vol. 11 (1977), no. 1, pp. 57–103.CrossRefGoogle Scholar
[5]Martin, D. A., Borel and projective determinacy, book to be published.Google Scholar
[6]Martin, D. A. and Steel, J. R., A proof of projective determinacy, Journal of the American Mathematical Society, vol. 2 (1989), pp. 71–125.CrossRefGoogle Scholar