Nowhere precipitousness of some ideals

Abstract

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).