Games with Filters

arXiv2009.04074MaRDI QIDQ6348718

Menachem Magidor, Matthew Foreman, Martin Zeman

Publication date: 8 September 2020

Abstract: This paper has two parts. The first is concerned with a variant of a family of games introduced by Holy and Schlicht, that we call emph{Welch games}. Player II having a winning strategy in the Welch game of length $omega$ on $kappa$ is equivalent to weak compactness. Winning the game of length $2^{k}appa$ is equivalent to $kappa$ being measurable. We show that for games of intermediate length $gamma$, II winning implies the existence of precipitous ideals with $gamma$-closed, $gamma$-dense trees. The second part shows the first is not vacuous. For each $gamma$ between $omega$ and $kapp{a}^{+}$, it gives a model where II wins the games of length $gamma$, but not $gamm{a}^{+}$. The technique also gives models where for all $omeg{a}_1<gammalekappa$ there are $kappa$-complete, normal, $kapp{a}^{+}$-distributive ideals having dense sets that are $gamma$-closed, but not $gamm{a}^{+}$-closed.

Mathematics Subject Classification ID

Large cardinals (03E55)

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