Bistationary Traces, Wide Levels, and Branch-Cover Rigidity for an Unrestricted Typed Variant of the Hayut-Magidor Forcing

Abstract

For every uncountable regular cardinal $\alpha$, ${\mathbb{S}}^{\ast }(\alpha )$ is an explicitly typed four-coordinate forcing motivated by the ladder-system construction of Hayut and Magidor. The forcing is $\sigma$-closed and, after adjoining a formal maximum, $\alpha$-strategically closed. For $\alpha \ge {\omega }_2$, every nonempty countable family of designated generic branches has a stationary and costationary common trace on the generic ladder-coordinate set $L_{\alpha }$, while no countable family of cofinal branches generates $L_{\alpha }$. These conclusions persist under a Kurepa-style level-size bound. In the unrestricted forcing, for every infinite cardinal $\mu <\alpha$ in the ground model, some level of the generic tree contains a copy of $({}^{\mu }2{)}^V$. Consequently, the endpoint-corrected restriction family indexed by ${\mathcal{P}}_{{\omega }_2}\alpha$ is too wide, whereas the scaled restriction system indexed by ${\mathcal{P}}_{\alpha }\alpha$ has all levels of size less than $\alpha$ exactly when $\alpha$ is strongly inaccessible in the ground model. When these equivalent conditions hold, the branch-covering number of $L_{\alpha }$ relative to the scaled system is at least ${\omega }_1$. The low-cofinality empty-value convention also ensures that the set of domains of $L_{\alpha }$ contains no club in ${\mathcal{P}}_{\alpha }\alpha$. The unrestricted tree clause of the motivating presentation is retained, without asserting forcing equivalence.

arXiv comment23 pages