Increasing the second uniform indiscernible by strongly ssp forcing

TL;DR

This paper introduces a new stationary set preserving forcing that increases the second uniform indiscernible under certain large cardinal assumptions, using novel game-theoretic techniques to control the forcing conditions.

Contribution

It presents a new forcing notion $ ext{P}^{c-c}$ that increases $ extbf{u}_2$ and develops open two-player games to analyze and control the forcing process.

Findings

The forcing $ ext{P}^{c-c}$ effectively increases $ extbf{u}_2$ beyond a given ordinal.

Introduction of capturing and catching-capturing games for analyzing forcing conditions.

Identification of special countable elementary submodels as side conditions.

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 $H_{\theta}^#$ 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.