**May 17**

Jonas Reitz,
CUNY

**Generalized Cohen Iterations**

Adding Cohen subsets to each of a class of cardinals in turn is a common construction in set theory, and underlies many fundamental results. The construction comes in two basic flavors, products (as in Eastonâ€™s Theorem on the powers of regular cardinals) and iterations (forcing the GCH). These flavors are apparently quite similar, forcing at stage kappa to add subsets via the Cohen partial order Add(kappa,lambda). They differ only in the universe over which Add(kappa,lambda) is defined - in the case of products the ground model poset is used at each stage, whereas in typical iterations the poset is taken from the partial extension up to kappa. In this talk I will consider an alternative, in which we allow Add(kappa,lambda) to be defined over an arbitrary inner model (lying between the ground model and the extension up to kappa) at each stage. These generalized Cohen iterations are ZFC-preserving, although neither the proof for products nor for traditional iterations transfers directly. They allow constructions such as class iterations of class products of Cohen forcing, with applications including new work with Kameryn Williams on iterating the Mantle.