Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-25T06:06:34.563Z Has data issue: false hasContentIssue false
  • English
  • Français

Ideals and combinatorial principles

Published online by Cambridge University Press:  12 March 2014

Douglas Burke  and
Yo Matsubara
Douglas Burke
Affiliation:
Department of Mathematical Science, University of Nevada, Las Vegas, Nevada 89154-4020, USA, E-mail: dburke@nevada.edu
Yo Matsubara
Affiliation:
School of Informatics and Sciences, Nagoya University, Nagoya, 461-01Japan, E-mail: yom@math.nagoya-u.ac.jp
Get access Rights & Permissions [Opens in a new window]

Extract

It is well known that if σ is a strongly compact cardinal and λ a regular cardinal ≥ σ, then for every stationary subset X of {α < λ: cof (α) = ω} there is some β < λ such that X ⋂ β is stationary in β. In fact the existence of a uniform, countably complete ultrafilter over λ is sufficient to prove the same conclusion about stationary subsets of {α < λ: cof (α) = ω}. See [13] or [10]. By analyzing the proof of this theorem as presented in [10], we realized the same conclusion will follow from the existence of a certain ideal, not necessarily prime, on . Throughout we will assume that σ is a regular uncountable cardinal and use the word “ideal” to mean fine ideal.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 62 , Issue 1 , March 1997 , pp. 117 - 122
Copyright
Copyright © Association for Symbolic Logic 1997

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]Abe, Y., Strong compactness and weakly normal ideals on , Fundamenta Mathematical, vol. 143 (1993), pp. 97–106.CrossRefGoogle Scholar
[2]Baumgartner, J. and Taylor, A., Saturation properties of ideals in generic extensions, II, Transactions of American Mathematical Society, vol. 271 (1982), pp. 587–609.Google Scholar
[3]Baumgartner, J., Taylor, A., and Wagon, S., Splitting stationary subsets of large cardinals, this Journal, vol. 42 (1977), pp. 203–214.Google Scholar
[4]Burke, D., Generic embedding and the failure of box, Proceedings of the AMS, vol. 123 (1995), pp. 2867–2871.CrossRefGoogle Scholar
[5]Burke, M. and Magidor, M., Shelah's pcf theory and it's application, Annals of Pure and Applied Logic, vol. 50 (1990), pp. 207–254.CrossRefGoogle Scholar
[6]Gitik, M., Nonsplitting subset of Pκ(κ +), this Journal, vol. 50 (1985), pp. 881–894.Google Scholar
[7]Goldring, N., Woodin cardinals and presaturated ideals, Annals of Pure and Applied Logic, vol. 55 (1992), pp. 285–303.CrossRefGoogle Scholar
[8]Jech, J., Set theory, Academic Press, New York, 1978.Google Scholar
[9]Johnson, C. A., On ideals and stationary reflection, this Journal, vol. 54 (1989), pp. 568–575.Google Scholar
[10]Kanamori, A. and Magidor, M., The evolution of large cardinal axioms in set theory, Higher set theory, Lecture Notes in Mathematics, vol. 669, Springer-Verlag, 1978.CrossRefGoogle Scholar
[11]Schimmerling, E., Combinatorial principles in the core model for one Woodin cardinal, Annals of Pure and Applied Logic, vol. 74 (1995), pp. 153–201.CrossRefGoogle Scholar
[12]Shelah, S., Proper forcing, Lecture Notes in Mathematics, vol. 940, Springer-Verlag.Google Scholar
[13]Solovay, R., Reinhardt, W., and Kanamori, A., Strong axioms of infinity and elementary embeddings, Annals of Mathematical Logic, vol. 13 (1978), pp. 73–116.CrossRefGoogle Scholar