On Subcomplete Forcing

TL;DR

This paper surveys subcomplete forcing, a novel class of forcing notions, exploring its properties, interactions with set-theoretic structures, and demonstrating that generalized diagonal Prikry forcing is subcomplete.

Contribution

It introduces and clarifies the concept of subcomplete forcing, comparing it to known forcing classes and analyzing its key properties and applications.

Findings

Subcomplete forcing preserves certain set-theoretic structures.

Generalized diagonal Prikry forcing is shown to be subcomplete.

Subcomplete forcing interacts with $\omega_1$-trees and maximality principles.

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 somewhat 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 $\omega_1$-trees, preservation properties of subcomplete forcing, the subcomplete maximality principle, the subcomplete resurrection axiom, and show that generalized diagonal Prikry forcing is subcomplete.