GCH的独立性和与Banach-Mazur对策有关的一个组合原理
The independence of GCH and a combinatorial principle related to Banach-Mazur games
摘要：
最近证明了关于Banach-Mazur博弈中部分信息策略的Telgársky猜想在$\mathsf{GCH}+\square$模型中是不成立的.该证明引入了一个从$\mathsf{GCH}+\square$中遵循的组合原理，即：$\triangledown$：每个具有$κ$ -cc的分离偏序集$\mathbb P$都包含一个稠密子集$\mathbb D$，使得对于每个$p \in \mathbb P$，都有$|\{ q \in \mathbb D \,:\, p \text{ extends } q \}| < κ$。我们证明了这个原则是独立的$\mathsf{GCH}$和$\mathsf{CH}$，在这个意义上，$\triangledown$不意味着$\mathsf{CH}$，和$\mathsf{GCH}$不意味着$\triangledown$假设一个巨大的基数的一致性。我们还考虑了更具体的问题，是否$\triangledown$持有与$\mathbb P$等于重量$\aleph_ω$措施代数。我们证明，再次假设一个巨大的基数的一致性，这个问题的答案是独立的$\mathsf{ZFC}+\mathsf{GCH}$。
英文摘要：
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 $\mathsf{ZFC}+\mathsf{GCH}$.
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