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On ideals and stationary reflection

Published online by Cambridge University Press:  12 March 2014

C. A. Johnson*
Affiliation:
Department of Mathematics, University of Keele, Keele, Staffordshire ST5 5BG, England

Extract

It is a theorem of Prikry [7] that if κ carries a uniform η-descendingly complete ultrafilter then the stationary reflection property fails. In this paper we will derive similar results, but here from properties of filters (or ideals) rather than ultrafilters.

Throughout κ and η will denote regular cardinals with η < κ (in particular κ will be uncountable), and I will denote an ideal on κ, by which we mean a set IP(κ) such that (i) I is closed under taking subsets and finite unions and (ii) αЄ I for each α < κ, but κI. I is said to be μ-complete if it is closed under taking unions of size < μ, I* = {XκκX Є I} is the filter dual to I and if A Є I+ (= P(κ) − I), then IA is the ideal on κ given by IA = {XκXA Є I}. If h: Aκ then h is said to be (i) unbounded mod I if for each α < κ, h−1(α) = {ξ Є Ah(ξ) < α} Є I and (ii) a least function for I if h is unbounded mod I and whenever g: Aκ is a function, unbounded mod I, then {ξ Є Ag{ξ) < h{ξ)} Є I.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1989

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References

REFERENCES

[1]Carr, D. M., The structure of ineffability properties of Pκλ (preprint).Google Scholar
[2]Galvin, F., Jech, T., and Magidor, M., An ideal game, this Journal, vol. 43 (1978), pp. 284291.Google Scholar
[3]Jech, T. and Prikry, K., Ideals over uncountable sets, Memoir no. 214, American Mathematical Society, Providence, Rhode Island, 1979.Google Scholar
[4]Kanamori, A., Weakly normal filters and irregular ultrafilters, Transactions of the American Mathematical Society, vol. 220 (1976), pp. 393399.CrossRefGoogle Scholar
[5]Kanamori, A. and Magidor, M., The evolution of large cardinal axioms in set theory, Higher set theory (proceedings, Oberwolfach, 1977), Lecture Notes in Mathematics, vol. 699, Springer-Verlag, Berlin, 1978, pp. 99275.CrossRefGoogle Scholar
[6]Komjath, P., Stationary reflection for uncountable cofinality, this Journal, vol. 51 (1986), pp. 147151.Google Scholar
[7]Prikry, K., On descendingly complete ultrafilters, Cambridge summer school in mathematical logic (1971), Lecture Notes in Mathematics, vol. 337, Springer-Verlag, Berlin, 1973, pp. 459488.CrossRefGoogle Scholar
[8]Solovay, R. M., Reinhardt, W. N., and Kanamori, A., Strong axioms of infinity and elementary embeddings, Annals of Mathematical Logic, vol. 13 (1978), pp. 73116.CrossRefGoogle Scholar
[9]Takahashi, J., A saturation property of ideals and weakly compact cardinals, this Journal, vol. 51 (1986), pp. 513525.Google Scholar