Date of Degree

6-2017

Document Type

Dissertation

Degree Name

Ph.D.

Program

Mathematics

Advisor

Gunter Fuchs

Committee Members

Joel David Hamkins

Arthur Apter

Keywords

Logic, Set Theory, Forcing

Abstract

I survey an array of topics in set theory in the context of a novel class of forcing notions: subcomplete forcing. Subcompleteness was originally defined by Ronald Jensen. I have attempted to make the subject more approachable to set theorists, while showing various properties of subcomplete forcing which one might desire of a forcing class, drawing comparisons between subcomplete forcing and countably closed forcing. In particular, I look at the interaction between subcomplete forcing and ω1-trees, preservation properties of subcomplete forcing, the subcomplete maximality principle, the subcomplete resurrection axiom, and show that diagonal Prikry forcing is subcomplete.

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