On Subcomplete Forcing

On Subcomplete Forcing by Kaethe Lynn Bruesselbach Minden Adviser: Professor Gunter Fuchs I survey an array of topics in set theory and their interaction with, or in the context of, a novel class of forcing notions: subcomplete forcing. Subcomplete forcing notions satisfy some desirable qualities; for example they don’t add any new reals to the model, and they admit an iteration theorem. While it is straightforward to show that any forcing notion that is countably closed is also subcomplete, it turns out that other well-known, more subtle forcing notions like Prikry forcing and Namba forcing are also subcomplete. Subcompleteness was originally defined by Ronald Björn Jensen around 2009. Jensen’s writings make up the vast majority of the literature on the subject. Indeed, the definition in and of itself is daunting. I have attempted to make the subject more approachable to set theorists, while showing various properties of subcomplete forcing that one might desire of a forcing class. It is well-known that countably closed forcings cannot add branches through ω1-trees. I look at the interaction between subcomplete forcing and ω1-trees. It turns out that subcomplete forcing also does not add cofinal branches to ω1-trees. I show that a myriad of other properties of trees of height ω1 as explored in [FH09] are preserved by subcomplete forcing; for example, I show that the unique branch property of Suslin trees is preserved by subcomplete forcing. Another topic I explored is the Maximality Principle (MP). Following in the footsteps of Hamkins [Ham03], Leibman [Lei], and Fuchs [Fuc08], [Fuc09], I examine the subcomplete maximality principle. In order to elucidate the ways in which subcomplete forcing generalizes