A general Mitchell style iteration

We work out the details of a schema for a mixed support forcing iteration, which generalizes the Mitchell model [7] with no Aronszajn trees on 2. The main purpose of this paper is to present the details of a schema for a mixed support forcing iteration. This schema will provide the technical framework for a variety of consistency results which we establish in [6] and [4]. In [5] we constructed a model in which the properties of being "internally club" and "internally approachable" are distinct, for structures of size the successor of a regular cardinal. Our construction was reminiscent in some ways of Mitchell's classic model with no Aronszajn trees on 2 ([7]). We used a forcing iteration whose factors are two-step iterations of Cohen forcing followed by a kind of collapse forcing, where the support on the Cohen part and on the collapsing part are of different sizes. However the kind of collapse forcing we used is not even strategically closed. In Section 2 we ...

• Type: Journal
• Year: 2008
• Where: MLQ
• Authors: John Krueger
• Keywords: Aronszajn Trees, Iteration, Mixed Support, MLQ 2008