We give an application of the extender based Radin forcing to cardinal arithmetic. Assuming κ is a large enough cardinal we construct a model satisfying 2κ = κ⁺ⁿ together with 2λ = λ⁺ⁿ for each cardinal λ < κ, where 0 < n < ω. The cofinality of κ can be set arbitrarily or κ can remain inaccessible. When κ remains an inaccessible, Vκ is a model of ZFC satisfying 2λ = λ+n for all cardinals λ.

The extender based Prikry forcing notion is being generalized to super compact extenders.

We show how to construct Gitik’s short extenders gap-3 forcing using a morass, and that the forcing notion is of Prikry type.

We show that Gitik’s short extender gap-2 forcing is of Prikry type.

We show that under certain large cardinal requirements there is a generic extension in which the power function behaves differently on different stationary classes. We achieve this by doing an Easton support iteration of the Radin on extenders forcing.