The stationary set splitting game

by Larson and Shelah. [LrSh:902]
Math Logic Quarterly, 2008
The emph {stationary set splitting game} is a game of perfect information of length omega_{1} 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 Sigma^{2}_{2} maximality with a predicate for the nonstationary ideal on omega_{1}, and an example of a consistently undetermined game of length omega_{1} 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.

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