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On ideals and stationary reflection

Published online by Cambridge University Press:  12 March 2014

C. A. Johnson
C. A. Johnson*
Affiliation:
Department of Mathematics, University of Keele, Keele, Staffordshire ST5 5BG, England
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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 I ⊆ P(κ) such that (i) I is closed under taking subsets and finite unions and (ii) αЄ I for each α < κ, but κ ∉ I. I is said to be μ-complete if it is closed under taking unions of size < μ, I* = {X ⊆ κ ∣ κ − X Є I} is the filter dual to I and if A Є I+ (= P(κ) − I), then I∣A is the ideal on κ given by I ∣ A = {X ⊆ κ ∣ X ∩ A Є I}. If h: A → κ then h is said to be (i) unbounded mod I if for each α < κ, h−1(α) = {ξ Є A ∣ h(ξ) < α} Є I and (ii) a least function for I if h is unbounded mod I and whenever g: A → κ is a function, unbounded mod I, then {ξ Є A ∣ g{ξ) < h{ξ)} Є I.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 54 , Issue 2 , June 1989 , pp. 568 - 575
Copyright
Copyright © Association for Symbolic Logic 1989

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References

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