Carlson Collapse is minimal under MA

Namba forcing [5] may be regarded as a generalization of Laver forcing [2] to ω2. The analogous forcing for ω1 we call the Carlson collapse. We first encountered it when writing our joint paper: Carlson, Kunen, and Miller [1]. In that paper we used the Prikry collapse of ω1, which is analogous to superperfect tree forcing (e.g., Miller [3]) but with subtrees of ω 1 . We proved in [1] the analogue of Theorem 3 for the Prikry collapse, namely that assuming Martin’s axiom the generic extension is minimal. Our paper [1] did not include Theorem 4, although Carlson had already proved it with an easier proof than is given here. Lemma 2 was obtained while giving a topics course [4] on forcing. One of my colleagues many years ago liked to joke about the referee report that said; “This paper fills a much needed gap in the literature.”