论文标题
GCH的独立性和与Banach-Mazur游戏有关的组合原则
The independence of GCH and a combinatorial principle related to Banach-Mazur games
论文作者
论文摘要
最近证明，Telgársky的猜想涉及Banach-Mazur游戏中的部分信息策略，在$ \ Mathsf {GCH}+\ Square $的型号中失败了。证明介绍了一个组合原理，该原理显示为$ \ mathsf {gch}+\ square $，即： $ \ triangledown $：$κ$ -CC包含一个密集的子poset $ \ mathbb d $，每个分离的poset $ \ mathbb p $，以至于$ | \ {q \ in \ mathbb d \，： <mathbb p $中的每$ p \ for <κ$。 We prove this principle is independent of $\mathsf{GCH}$ and $\mathsf{CH}$, in the sense that $\triangledown$ does not imply $\mathsf{CH}$, and $\mathsf{GCH}$ does not imply $\triangledown$ assuming the consistency of a huge cardinal. 我们还考虑了$ \ triangledown $用$ \ mathbb p $等于重量的更具体的问题 - $ \aleph_Ω$ measure elgebra。我们再次证明，假设一个巨大的红衣主教的一致性，该问题的答案独立于$ \ mathsf {zfc}+\ \ \ \ mathsf {gch} $。
It was proved recently that Telgársky's conjecture, which concerns partial information strategies in the Banach-Mazur game, fails in models of $\mathsf{GCH}+\square$. The proof introduces a combinatorial principle that is shown to follow from $\mathsf{GCH}+\square$, namely: $\triangledown$: Every separative poset $\mathbb P$ with the $κ$-cc contains a dense sub-poset $\mathbb D$ such that $|\{ q \in \mathbb D \,:\, p \text{ extends } q \}| < κ$ for every $p \in \mathbb P$. We prove this principle is independent of $\mathsf{GCH}$ and $\mathsf{CH}$, in the sense that $\triangledown$ does not imply $\mathsf{CH}$, and $\mathsf{GCH}$ does not imply $\triangledown$ assuming the consistency of a huge cardinal. We also consider the more specific question of whether $\triangledown$ holds with $\mathbb P$ equal to the weight-$\aleph_ω$ measure algebra. We prove, again assuming the consistency of a huge cardinal, that the answer to this question is independent of $\mathsf{ZFC}+\mathsf{GCH}$.
