Hayut-Magidor力迫的无限制类型变体的双平稳迹、宽层与分支覆盖刚性
Bistationary Traces, Wide Levels, and Branch-Cover Rigidity for an Unrestricted Typed Variant of the Hayut-Magidor Forcing
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- School of Mathematics and Statistics, Hanjiang Normal University（汉江师范学院数学与统计学院）
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中文总结 AI 辅助
本文研究Hayut-Magidor力迫的无限制类型变体，证明其泛型梯子坐标集的平稳迹性质、层宽条件与分支覆盖刚性，并给出强不可达基数下的等价刻画。
中文摘要 AI 辅助
对于每一个不可数正则基数 $\alpha$，$\mathbb S^{\ast}(\alpha)$ 是一个显式类型的四坐标力迫，其动机来自Hayut和Magidor的梯子系统构造。该力迫是 $\sigma$-闭的，并且在添加一个形式最大元后，是 $\alpha$-策略闭的。对于 $\alpha\geq\omega_2$，每个非空可数族指定的泛型分支在泛型梯子坐标集 $L_\alpha$ 上都有一个平稳且余平稳的共同迹，而没有可数族共尾分支生成 $L_\alpha$。这些结论在Kurepa式的层大小界下仍然成立。在无限制力迫中，对于基模型中每个无穷基数 $\mu<\alpha$，泛型树的某一层包含 $({}^\mu 2)^V$ 的一个副本。因此，由 $\mathcal P_{\omega_2}\alpha$ 索引的端点修正限制族过宽，而由 $\mathcal P_\alpha\alpha$ 索引的缩放限制系统恰好当 $\alpha$ 在基模型中强不可达时，其所有层的大小都小于 $\alpha$。当这些等价条件成立时，$L_\alpha$ 相对于缩放系统的分支覆盖数至少为 $\omega_1$。低共尾性空值约定还确保 $L_\alpha$ 的定义域集合在 $\mathcal P_\alpha\alpha$ 中不包含俱乐部。动机呈现的无限制树子句被保留，但不断言力迫等价。
英文摘要
For every uncountable regular cardinal $α$, $\mathbb S^{\ast}(α)$ is an explicitly typed four-coordinate forcing motivated by the ladder-system construction of Hayut and Magidor. The forcing is $σ$-closed and, after adjoining a formal maximum, $α$-strategically closed. For $α\geqω_2$, every nonempty countable family of designated generic branches has a stationary and costationary common trace on the generic ladder-coordinate set $L_α$, while no countable family of cofinal branches generates $L_α$. These conclusions persist under a Kurepa-style level-size bound. In the unrestricted forcing, for every infinite cardinal $μ<α$ in the ground model, some level of the generic tree contains a copy of $({}^μ2)^V$. Consequently, the endpoint-corrected restriction family indexed by $\mathcal P_{ω_2}α$ is too wide, whereas the scaled restriction system indexed by $\mathcal P_αα$ has all levels of size less than $α$ exactly when $α$ is strongly inaccessible in the ground model. When these equivalent conditions hold, the branch-covering number of $L_α$ relative to the scaled system is at least $ω_1$. The low-cofinality empty-value convention also ensures that the set of domains of $L_α$ contains no club in $\mathcal P_αα$. The unrestricted tree clause of the motivating presentation is retained, without asserting forcing equivalence.
