Increasing the second uniform indiscernible by strongly ssp forcing

Abstract

We introduce a new and natural stationary set preserving forcing ${\mathbb{P}}^{c-c}(\lambda ,\mu )$ that (under ${\mathsf{NS}}_{{\omega }_1}$ precipitous + existence of $H_θ^#$ for a sufficiently large regular $\theta$) increases the second uniform indiscernible ${\mathbf{u}}_2$ beyond some given ordinal $\lambda$. 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 $M\prec H_{\theta }$ that occur as side conditions in ${\mathbb{P}}^{c-c}$ and thus allow to control the forcing in a strong way.

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