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2. Games

    1.     Let \kappa be a regular cardinal and \gamma an infinite ordinal. Consider the following two-player game (called the Welch game): Player I plays \kappa-algebras \mathscr{A}_i, while Player II plays increasing \kappa-complete filters \mathcal{F}_i on \mathscr{A}_i. The game goes on for \gamma many steps. The first player to break one of the rules loses.

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      Problem 2.1.

      [Foreman]
      1. For what values of \kappa and \gamma are these games determined?
      2. How can these games be generalized to, for example, extender sequences? What about strong compactness and supercompactness?
          Known facts:

      1. (Keisler-Tarksi) If \gamma=\omega, then II has a winning strategy.

      2. (Foreman-Magidor-Shelah) If II has a winning strategy when \gamma=\omega+1, then there is a precipitous ideal on \kappa.

      3. Suppose GCH holds and \omega_1<\text{cf}(\gamma)=\gamma<\kappa^+. If II has a winning strategy, then there exists a \kappa-complete ideal on \kappa which has a dense tree T of height \gamma which is closed under descending sequences of length <\gamma.

      4. If \kappa is measurable, then II has a winning strategy in the game of length 2^\kappa.

          Cite this as: AimPL: From ℵ2 to infinity, available at http://aimpl.org/alephtwo.