6 Fe b 19 93 Combinatorics on Ideals and Axiom A

The following notion of forcing was introduced by Grigorieff [2]: Let I ⊂ ω be an ideal, then P is the set of all functions p : ω → 2 such that dom(p) ∈ I. The usual Cohen forcing corresponds to the case when I is the ideal of finite subsets of ω. In [2] Grigorieff proves that if I is the dual of a p-point ultrafilter, then ω1 is preserved in the generic extension. Later, when Shelah introduced the notion of proper forcing, many people observed that Grigorieff forcing was proper. One way of proving this is to show that player II has a winning strategy in the game Gω for P (see [3], page 91.) The notion of Axiom A forcing was introduced by Baumgartner [1]. If P satisfies Axiom A, then player II has a winning strategy in the game Gω and thus is proper. Indeed, most of the naturally occurring proper notions of forcing are Axiom A (e.g. Mathias or Laver forcing). Thus it is natural to ask whether or not Grigorieff forcing satisfies Axiom A. The main result of this paper is a negative answer to this question. We will prove this by introducing another game GU and showing that if P were Axiom A then player II would have a winning strategy in this game. We will then prove that the game GU is undetermined.