Diagonal Supercompact Radin Forcing

In this paper we consider two methods for producing models with some global behavior of the continuum function on singular cardinals and the failure of weak square. The first method is as an extension of Sinapova’s work [21]. We define a diagonal supercompact Radin forcing which adds a club subset to a cardinal κ while forcing the failure of SCH everywhere on the club. The intuition from Sinapova’s work was that our model should have the failure of weak square at every point in the club. Unfortunately, the additional closure required to preserve inaccessibility enforces weak square at some points.