**March 31**

Benjamin Goodman,
CUNY

**$\Sigma_n$-correct forcing axioms**

The standard method of producing a model of a forcing axiom from a supercompact cardinal in fact gives a model of an even stronger principle: that for every small name a and every $\Sigma_2$ formula $arphi$ such that $\varphi(a)$ is forceable by and preserved under further forcing in our forcing class, there is a filter $F$ which meets a desired collection of dense sets and also interprets a such that $\varphi(a^F)$ already holds. I will show how to generalize this result to formulas of higher complexity by starting with slightly stronger large cardinal assumptions, then discuss the bounded versions of these enhanced forcing axioms, their relationships to other similar principles, and their consequences.