Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-16T13:21:44.616Z Has data issue: false hasContentIssue false
  • English
  • Français

Nowhere precipitousness of some ideals

Published online by Cambridge University Press:  12 March 2014

Yo Matsubara  and
Masahiro Shioya
Yo Matsubara
Affiliation:
School of Informatics and Sciences, Nagoya University, Nagoya, 464, Japan E-mail: yom@math.nagoya-u.ac.jp
Masahiro Shioya
Affiliation:
Institute of Mathematics, University of Tsukuba, Tsukuba, 305, Japan E-mail: shioya@math.cc.tsukuba.ac.jp
Get access Rights & Permissions [Opens in a new window]

Extract

In this paper we will present a simple condition for an ideal to be nowhere precipitous. Through this condition we show nowhere precipitousness of fundamental ideals on Pkλ, in particular the non-stationary ideal NSkλ under cardinal arithmetic assumptions.

In this section I denotes a non-principal ideal on an infinite set A. Let I+ = PA / I (ordered by inclusion as a forcing notion) and I∣X = {Y ⊂ A: Y ⋂ X ∈ I}, which is also an ideal on A for X ∈ I+. We refer the reader to [8, Section 35] for the general theory of generic ultrapowers associated with an ideal. We recall I is said to be precipitous if ⊨I+ “Ult(V, Ġ) is well-founded” [9].

The central notion of this paper is a strong negation of precipitousness [1]:

Definition. I is nowhere precipitous if I∣X is not precipitous for every X ∈ I+ i.e., ⊨I+ “Ult(V, Ġ) is ill-founded.”

It is useful to characterize nowhere precipitousness in terms of infinite games (see [11, Section 27]). Consider the following game G(I) between two players, Nonempty and Empty [5]. Nonempty and Empty alternately choose Xn ∈ I+ and Yn ∈ I+ respectively so that Xn ⊃ Yn ⊃n+1. After ω moves, Empty wins the game if⋂n<ωXn=⋂n<ωYn = Ø.

See [5, Theorem 2] for a proof of the following characterization.

Proposition. I is nowhere precipitous if and only if Empty has a winning strategy in G(I).

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 63 , Issue 3 , September 1998 , pp. 1003 - 1006
Copyright
Copyright © Association for Symbolic Logic 1998

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]Baumgartner, J. and Taylor, A., Saturation properties of ideals in generic extensions II, Transactions of the American Mathematical Society, vol. 271 (1982), pp. 587–609.Google Scholar
[2]Burke, M. and Magidor, M., Shelah's pcf theory and its applications, Annals of Pure and Applied Logic, vol. 50 (1990), pp. 207–254.CrossRefGoogle Scholar
[3]Donder, H.-D. and Matet, P., Two cardinal versions of diamond, Israel Journal of Mathematics, vol. 83 (1993), pp. 1–43.CrossRefGoogle Scholar
[4]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
[5]Galvin, F., Jech, T., and Magidor, M., An ideal game, this Journal, vol. 43 (1978), pp. 284–292.Google Scholar
[6]Goldring, N., preprint.Google Scholar
[7]Galvin, F., Projecting precipitousness, Israel Journal of Mathematics, vol. 74 (1991), pp. 13–31.Google Scholar
[8]Jech, T., Set theory, Academic Press, 1978.Google Scholar
[9]Jech, T. and Prikry, K., Ideals over uncountable sets: Application of almost disjoint functions and generic ultrapowers, Memoirs of the American Mathematical Society, vol. 214 (1979).Google Scholar
[10]Johnson, C., Seminormal λ-generated ideals on P kλ, this Journal, vol. 53 (1988), pp. 92–102.Google Scholar
[11]Kanamori, A., The higher infinite, Springer-Verlag, 1994.Google Scholar
[12]Shelah, S., ℵω+1 has a Jonsson algebra, Cardinal arithmetic, Oxford University Press, 1994, pp. 34–116.CrossRefGoogle Scholar