Iteration theorems for subversions of forcing classes

TL;DR

This paper establishes iteration theorems for classes of subproper and subcomplete forcing, demonstrating their preservation properties using revised countable support and nice iterations, advancing the understanding of forcing class iterability.

Contribution

It introduces new iteration theorems for subproper and subcomplete forcing classes, utilizing revised countable support and nice iterations, and simplifies their definitions while maintaining key properties.

Findings

Subproper, ${}^ ext{omega} ext{omega}$-bounding forcing notions are iterable with revised countable support.

Subproper, $T$-preserving and $[T]$-preserving forcing notions are also iterable with revised countable support.

Nice iterations allow dropping technical conditions, still preserving $ ext{omega}_1$ and other properties.

Abstract

We prove various iteration theorems for forcing classes related to subproper and subcomplete forcing, introduced by Jensen. In the first part, we use revised countable support iterations, and show that 1) the class of subproper, ${}^\omega\omega$-bounding forcing notions, 2) the class of subproper, $T$-preserving forcing notions (where $T$ is a fixed Souslin tree) and 3) the class of subproper, $[T]$-preserving forcing notions (where $T$ is an $\omega_1$-tree) are iterable with revised countable support. In the second part, we adopt Miyamoto's theory of nice iterations, rather than revised countable support. We show that this approach allows us to drop a technical condition in the definitions of subcompleteness and subproperness, still resulting in forcing classes that are iterable in this way, preserve $\omega_1$, and, in the case of subcompleteness, don't add reals. Further, we show that the analogs of the iteration theorems proved in the first part for RCS iterations hold for nice iterations as well.