Combinatorics on Ideals and Axiom A

Abstract

Throughout this abstract let U be a fixed p-point ultrafilter and let I be the dual ideal. Grigorieff forcing is P(U)={p:omega to 2|dom(p) is an element of I} ordered by reverse inclusion. It is well known that Grigorieff forcing is proper. The main result of this paper is the following: THEOREM: Gregorieff forcing does not satisfy Axiom A. To prove this we use the following game, denoted G(U), for two players playing alternatively: Player I plays a partition of omega, {J_n| n<omega}, such that for all n<omega, J_n is an element of I; At the nth turn Player II plays a finite subset F_n of J_n. Player II wins iff the union of the F_n is an element of U. The following two Lemmas prove the Theorem: LEMMA 1: If P(U) satisfies axiom A, then player II has a winning strategy in the game G(U). LEMMA 2:The game G(U) is undetermined.