8 - Short extender forcing

Summary

The eleventh Appalachian Set Theory workshop was held at Carnegie Mellon University on April 3, 2010. The lecturer was Moti Gitik. As a graduate student Spencer Unger assisted in writing this chapter, which is based on the workshop lectures.

Introduction

The goal of these notes is to provide the reader with an introduction to the main ideas of a result due to Gitik [2].

Theorem 1.1 Let 〈κ_n ∣ n < ω〉 be an increasing sequence with each -strong, and κ =_{de f} sup_n<ω κ_n. There is a cardinal preserving forcing extension in which no bounded subsets of κ are added and κ^ω = κ⁺⁺.

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 〈κ_n ∣ n < ω〉 with κ =_{de f} sup_n<ω κ_n. 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.