### 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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