The Stationary Set Splitting Game

Details The Stationary Set Splitting Game The Stationary Set Splitting Game

2010-03-11

arxiv.org/abs/1003.2425v1

Mathematical Logic Quarterly 54 (2008) 2, 187-193

Mathematics

Logic

Scientific paper

The \emph{stationary set splitting game} is a game of perfect information of length $\omega_{1}$ between two players, \unspls and \spl, in which \unspls chooses stationarily many countable ordinals and \spls 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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