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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
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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

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References

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