Many countable support iterations of proper forcings preserve Souslin trees

TL;DR

This paper demonstrates that a broad class of countable support iterations of proper forcings can preserve Souslin trees, providing new conditions and proofs that unify and extend existing preservation results.

Contribution

It introduces new sufficient conditions for preservation using game-theoretic approaches and offers a unified proof method applicable to all cases without dividing forcings.

Findings

Countable support iterations of proper forcings preserve Souslin trees under new conditions.

Connections established between preservation properties and game-theoretic conditions.

A proof technique that does not require dividing forcings into those adding reals and those that do not.

Abstract

We show that many countable support iterations of proper forcings preserve Souslin trees. We establish sufficient conditions in terms of games and we draw connections to other preservation properties. We present a proof of preservation properties in countable support interations in the so-called Case A that does not need a division into forcings that add reals and those who do not.