The Stationary Set Splitting Game

TL;DR

This paper investigates the stationary set splitting game of length 1, showing that in ZFC either player can have a winning strategy or the game can be undetermined, with implications for set theory and logic.

Contribution

It provides the first example of a 1-length game with a definable payoff that can be undetermined in ZFC, and explores its consistency with various set-theoretic axioms.

Findings

It is consistent in ZFC that either player has a winning strategy or neither does.

The game is an example of a 1-length game with a definable payoff that can be undetermined.

Determinacy of the game is consistent with Martin's Axiom but not with Martin's Maximum.

Abstract

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.