Skip to main content Accessibility help

Ideals and combinatorial principles

  • Douglas Burke (a1) and Yo Matsubara (a2)


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 Xβ 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.



Hide All
[1]Abe, Y., Strong compactness and weakly normal ideals on , Fundamenta Mathematical, vol. 143 (1993), pp. 97106.
[2]Baumgartner, J. and Taylor, A., Saturation properties of ideals in generic extensions, II, Transactions of American Mathematical Society, vol. 271 (1982), pp. 587609.
[3]Baumgartner, J., Taylor, A., and Wagon, S., Splitting stationary subsets of large cardinals, this Journal, vol. 42 (1977), pp. 203214.
[4]Burke, D., Generic embedding and the failure of box, Proceedings of the AMS, vol. 123 (1995), pp. 28672871.
[5]Burke, M. and Magidor, M., Shelah's pcf theory and it's application, Annals of Pure and Applied Logic, vol. 50 (1990), pp. 207254.
[6]Gitik, M., Nonsplitting subset of Pκ(κ +), this Journal, vol. 50 (1985), pp. 881894.
[7]Goldring, N., Woodin cardinals and presaturated ideals, Annals of Pure and Applied Logic, vol. 55 (1992), pp. 285303.
[8]Jech, J., Set theory, Academic Press, New York, 1978.
[9]Johnson, C. A., On ideals and stationary reflection, this Journal, vol. 54 (1989), pp. 568575.
[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.
[11]Schimmerling, E., Combinatorial principles in the core model for one Woodin cardinal, Annals of Pure and Applied Logic, vol. 74 (1995), pp. 153201.
[12]Shelah, S., Proper forcing, Lecture Notes in Mathematics, vol. 940, Springer-Verlag.
[13]Solovay, R., Reinhardt, W., and Kanamori, A., Strong axioms of infinity and elementary embeddings, Annals of Mathematical Logic, vol. 13 (1978), pp. 73116.

Ideals and combinatorial principles

  • Douglas Burke (a1) and Yo Matsubara (a2)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed