**May 4**

Jonas Reitz,
CUNY

**Cohen Forcing and Inner Models**

The most well-known method for adding a subset to a regular cardinal kappa over the universe $V$ is the Cohen partial order ${\rm Add}(\kappa,1)$, whose conditions consist of binary sequences bounded in $\kappa$ and ordered by end extension. Presented with an inner model $W$, however, we can consider an alternative: add a subset to kappa over the universe $V$ using ${\rm Add}(\kappa,1)^W$, the Cohen partial order as defined within $W$. This situation is not unusual in set theory - it arises in the canonical methods for adding subsets to multiple cardinals (in product forcing we always use the poset of the ground model $W$, whereas in iterations we use the poset of the extension), and in many other inner and outer model constructions. As $W$ may have fewer bounded subsets of kappa than $V$, these posets may not be equal. How do they compare?

In this talk I will analyze ${\rm Add}(\kappa,1)^W$ and ${\rm Add}(\kappa,1)^V$ with regards to their relative forcing strength. I will offer a complete characterization of when forcing with ${\rm Add}(\kappa,1)^W$ is (at least) as strong as ${\rm Add}(\kappa,1)^V$, in the sense that forcing with the former poset adds a generic for the latter. I will present partial results in the reverse direction (when is ${\rm Add}(\kappa,1)^V$ as strong as ${\rm Add}(\kappa,1)^W$?), and discuss open questions and applications.