Almost Strong Properness

TL;DR

This paper introduces the concept of almost strong properness, a new forcing property that lies between properness and strong properness, and applies it to force the MRP using a simple finite-condition forcing.

Contribution

It defines almost strong properness and demonstrates its application in forcing the MRP with a straightforward finite-condition forcing.

Findings

Introduces the forcing property 'almost strong properness'

Establishes the position of this property between properness and strong properness

Provides a simple forcing construction to force MRP

Abstract

We introduce the forcing property "almost strong properness" which sits between properness and strong properness. As an application, we introduce a simple forcing with finite conditions to force $\rm MRP$.