The stationary set splitting game

The stationary set splitting game is a game of perfect information of length ω1 between two players, unsplit and split, in which unsplit chooses stationarily many countable ordinals and split tries to continuously divide them into two stationary pieces. We show that it is possible in ZFC to force a winning strategy for either player, or for neither. This gives a new counterexample to Σ2 maximality with a predicate for the nonstationary ideal on ω1, and an example of a consistently undetermined game of length ω1 with payoff definable in the second-order monadic logic of order. We also show that the determinacy of the game is consistent with Martin’s Axiom but not Martin’s Maximum. MSC2000: 03E35; 03E60 The stationary set splitting game (SG) is a game of perfect information of length ω1 between two players, unsplit and split. In each round α, unsplit either accepts or rejects α. If unsplit accepts α, then split puts α into one of two sets A and B. If unsplit rejects α then split does nothing. After all ω1 many rounds have been played, split wins if unsplit has not accepted stationarily often, or if both of A and B are stationary. In this note we prove that it is possible to force a winning strategy for either player in SG, or for neither, and we also show that the determinacy of SG is consistent with Martin’s Axiom but not Martin’s Maximum [4]. We also present two guessing principles, Cs (club for split) and Du (diamond for unsplit), which imply the existence of winning strategies for split and unsplit, respectively (and are therefore incompatible; see Theorems 1.5 and 1.8). These principles may be of independent interest. ∗The work in this paper began during the Set Theory and Analysis program at the Fields Institute in the Fall of 2002. The first author is supported in part by NSF grant DMS-0401603, and thanks Juris Steprāns, Paul Szeptycki and Tetsuya Ishiu for helpful conversations on this topic. The research of the second author is supported by the United States-Israel Binational Science Foundation. This is the second author’s publication 902. Some of the research in this paper was conducted during a visit by the first author to Rutgers University, supported by NSF grant DMS-0600940.