Résume | In order to make this talk accessible for a general audience of logicians, we will start off softly by first introducing certain open two-player games and then use these to isolate a special family of countable elementary submodels of the structure H_\theta. We will then analyse the existence of (projective stationary) many such models and discuss the connection with precipitousness of the nonstationary ideal on \omega_1. Next, we will discuss a particular forcing P that consists of finite conditions in which these special models feature as side conditions. Depending on the time, we might go on to survey some interesting properties that this forcing shares with an L-forcing defined by Claverie Schindler and a Namba-like forcing defined by Ketchersid-Larson-Zapletal, to both of which it shows resemblance. It will follow that this side condition approach using special models gives yet another way to increase the second uniform indiscernible in a stationary set preserving way beyond some arbitrary prespecified ordinal. This is all part of joint ongoing work with my thesis supervisor Boban Velickovic. |