Short Extender Forcing

In order to present this result, we approach it by proving some preliminary theorems about different forcings which capture the main ideas in a simpler setting. For the entirety of the notes we work with an increasing sequence of large cardinals 〈κn | n < ω〉 with κ =def supn<ω κn. The large cardinal hypothesis that we use varies with the forcing. A recurring theme is the idea of a cell. A cell is a simple poset which is designed to be used together with other cells to form a large poset. Each of the forcings that we present has ω-many cells which are put together in a canonical way to make the forcing. First, we will present Diagonal Prikry Forcing which adds a single cofinal ωsequence to κ. The key property that we wish to present with this forcing is the Prikry condition. The Prikry condition is the property of the forcing that allows us to show that no bounded subsets of κ are added. All subsequent forcings share this property. We also show that this forcing preserves cardinals and cofinalities above κ using a chain condition argument. Second, we present a forcing for adding λ-many ω-sequences to κ using long extenders. This forcing can be seen as both a more complicated version of Diagonal Prikry forcing and an approximation of the forcing used to prove Theorem 1. We want to repeat many of the arguments from the first poset. Each argument becomes more difficult. We sketch a proof of the Prikry condition and mention a strengthening needed to show that κ is preserved. The chain condition argument is also more difficult and we sketch a proof of this too. Third, we present the forcing from Theorem 1. To do this we present an attempt at a definition, which we ultimately refine to obtain the actual forcing. This is instructive because our attempt at a definition is similar to the forcing with long