March 8
Brent Cody,
Virginia Commonwealth University
Generic embeddings and ideals associated to large cardinals
Generic elementary embeddings have many applications throughout set theory. For example, Solovay's original proof that every stationary subset of a regular uncountable cardinal $\kappa$ can be partitioned into $\kappa$ disjoint stationary sets involves the use of a generic ultrapower embedding obtained by forcing with the nonstationary ideal. The generic ultrapower construction leads naturally to a characterization of the nonstationary ideal on a regular uncountable cardinal $\textrm{NS}_\kappa$ in terms of generic elementary embeddings. We will show that this characterization can be generalized to various ideals associated to large cardinals. Specifically, we will discuss characterizations of the $\Pi^m_n$-indescribability ideals, the subtle ideal and the ineffability ideal in terms of generic elementary embeddings as well as potential applications.