Gitik’s Gap 2 Short Extender Forcing

In an Appalachian Set Theory Workshop [3], Gitik presented some of the details of a simplified version of the poset from his original paper. The discussion there motivates the definition of the forcing by modifying a poset which requires a stronger large cardinal assumption which is sometimes called the long extender forcing. However, the discussion recorded in the Appalachian Set Theory (AST) notes falls short of giving a complete account of this simplified forcing. The purpose of this paper is to fill in the gaps from the AST notes and so give a complete account of this simplified forcing. In particular we focus the proof that the final forcing has the κ-cc. Since the motivation for the poset is nicely outlined in the AST notes we jump straight into the technical details. One note on convention is in order. We depart from previous presentations of this poset by using p ≤ q to mean that p is stronger than q.