论文标题
通过强烈的SSP强迫增加不可分割的第二均均匀
Increasing the second uniform indiscernible by strongly ssp forcing
论文作者
论文摘要
我们介绍了一套新的天然平稳套件，以防止强迫$ \ mathbb p^{c-c}（λ，μ）$（在$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ m artsf {ns} _ {ω_1} $ stragitous + stragitous +存在$h_θ^＃$ $ $h_θ^＃$的存在中，超过了第二个常规$θ$），超过了一定的$θ$）序数$λ$。强迫$ \ mathbb p^{c-c} $与[2]和[9]中定义的强制性共享此属性。作为主要工具，我们使用某些具有独立兴趣的自然开放式玩家游戏，即。捕获游戏$ \ mathbf {g} _m^{cap}（x）$和捕获捕获的游戏$ \ mathbf {g} _m^{c-c}（c-c}（x）$。特别是，这些游戏用于隔离一个特殊的可数基本subsodels $ m \ prech_θ$的特殊家族，该家族以$ \ mathbb p^{c-c} $中的侧面条件发生，因此允许以强大的方式控制强迫。
We introduce a new and natural stationary set preserving forcing $\mathbb P^{c-c}(λ,μ)$ that (under $\mathsf{NS}_{ω_1}$ precipitous + existence of $H_θ^#$ for a sufficiently large regular $θ$) increases the second uniform indiscernible $\mathbf{u}_2$ beyond some given ordinal $λ$. The forcing $\mathbb P^{c-c}$ shares this property with forcings defined in [2] and [9]. As a main tool we use certain natural open two player games which are of independent interest, viz. the capturing games $\mathbf{G}_M^{cap}(X)$ and the catching-capturing games $\mathbf{G}_M^{c-c}(X)$. In particular, these games are used to isolate a special family of countable elementary submodels $M \prec H_θ$ that occur as side conditions in $\mathbb P^{c-c}$ and thus allow to control the forcing in a strong way.
