Subcomplete forcing, trees and generic absoluteness

TL;DR

This paper studies how subcomplete forcing affects trees of height , showing it cannot add branches to -trees and analyzing preservation of Aronszajn trees under various set-theoretic assumptions.

Contribution

It introduces fragments of subcompleteness preserved by subcomplete forcing and links preservation of trees to bounded forcing axioms and generic absoluteness.

Findings

Subcomplete forcing cannot add new branches to -trees.

Preservation of Aronszajn trees depends on CH and bounded subcomplete forcing axiom.

Certain rigidity properties of Suslin trees are preserved under subcomplete forcing.

Abstract

We investigate properties of trees of height $\omega_1$ and their preservation under subcomplete forcing. We show that subcomplete forcing cannot add a new branch to an $\omega_1$-tree. We introduce fragments of subcompleteness which are preserved by subcomplete forcing, and use these in order to show that certain strong forms of rigidity of Suslin trees are preserved by subcomplete forcings. Finally, we explore under what circumstances subcomplete forcing preserves Aronszajn trees of height and width $\omega_1$. We show that this is the case if CH fails, and if CH holds, then this is the case iff the bounded subcomplete forcing axiom holds. Finally, we explore the relationships between bounded forcing axioms, preservation of Aronszajn trees of height and width $\omega_1$ and generic absoluteness of $\Sigma^1_1$-statements over first order structures of size $\omega_1$, also for other canonical classes of forcing.