Preservation of a Souslin tree and side conditions

TL;DR

This paper introduces a method to force a specific forcing axiom related to proper forcing notions that preserve a given Souslin tree, using finite conditions and a specialized iteration with side conditions.

Contribution

It develops a Neeman style iteration with generalized side conditions to establish the consistency of PFA(T), a relativized forcing axiom, with preservation of a Souslin tree.

Findings

Established a new forcing technique with finite conditions.

Proved the consistency of PFA(T) with a preserved Souslin tree.

Extended preservation theorems for complex iterations.

Abstract

We show how to force, with finite conditions, the forcing axiom PFA(T), a relativization of PFA to proper forcing notions preserving a given Souslin tree T. The proof uses a Neeman style iteration with generalized side conditions consisting of models of two types, and a preservation theorem for such iterations. The consistency of this axiom was previously known by the standard countable support iteration, using a preservation theorem due to Miyamoto.