  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.
