Coherent Adequate Sets and Forcing Square

TL;DR

This paper introduces coherent adequate sets of models as side conditions in forcing, and uses them to define a forcing that adds a square sequence on a9_2 with finite conditions.

Contribution

It presents a new concept of coherent adequate sets and applies it to construct a forcing for adding a square sequence on a9_2.

Findings

Defined a new class of models called coherent adequate sets.

Constructed a forcing poset with finite conditions that adds a square sequence.

Demonstrated the effectiveness of coherent adequate sets in forcing constructions.

Abstract

We introduce the idea of a coherent adequate set of models, which can be used as side conditions in forcing. As an application we define a forcing poset which adds a square sequence on $\omega_2$ using finite conditions.