论文标题
将基础迫使$ \ aleph_1 $ - free组
Forcing a Basis into $\aleph_1$-Free Groups
论文作者
论文摘要
在本文中，我们解决了一个问题，即何时可以在瞬态的基数传播模型扩展名中免费免费$ \ aleph_1 $ $ h $ $ h $。使用$γ$ -Invariant，表示$γ（H）$，我们提出了一个必要且充分的条件，可以解决此问题，以$ \ aleph_1 $ -free -free基团的基数$ \ Aleph_1 $。具体而言，如果$γ（h）= [\ aleph_1] $，则$ h $在且仅当$ \ aleph_1 $倒塌时在传递模型扩展中是免费的，而对于$γ（h）\ ne [\ aleph_1] $ collapses collapses collapses n时，存在基于基础的基础，将为$ h $ h $ h $ h $ h $ h $添加基础。特别是，对于$γ（h）\ neq [\ aleph_1] $，我们提供了Poset $（\ Mathcal p _ {\ rm pb}，\ rm pb}，\ leq），\ leq）$ h $添加基础的部分基础，而不会折叠$ \ aleph_1 $。
In this paper, we address the question of when a non-free $\aleph_1$-free group $H$ can be be free in a transitive cardinality-preserving model extension. Using the $Γ$-invariant, denoted $Γ(H)$, we present a necessary and sufficient condition resolving this question for $\aleph_1$-free groups of cardinality $\aleph_1$. Specifically, if $Γ(H) = [\aleph_1]$, then $H$ will be free in a transitive model extension if and only if $\aleph_1$ collapses, while for $Γ(H) \ne [\aleph_1]$ there exist cardinality-preserving forcings that will add a basis to $H$. In particular, for $Γ(H) \neq [\aleph_1]$, we provide a poset $(\mathcal P_{\rm pb}, \leq)$ of partial bases for adding a basis to $H$ without collapsing $\aleph_1$.
