Morasses and finite support iterations

TL;DR

This paper presents a novel method for constructing forcing notions along simplified morasses that satisfy the chain condition, enabling the creation of specific combinatorial structures like an -Suslin tree.

Contribution

It introduces a new technique to build forcing along morasses ensuring the -chain condition, expanding tools for set-theoretic constructions.

Findings

Constructed a -cc forcing adding an -Suslin tree.

Developed a method to thin out larger forcings to satisfy chain conditions.

Connected the method to existing approaches like Shelah's historic forcing.

Abstract

We introduce a method of constructing a forcing along a simplified $(\kappa,1)$-morass such that the forcing satisfies the $\kappa$-chain condition. Alternatively, this may be seen as a method to thin out a larger forcing to get a chain condition. As an application, we construct a ccc forcing that adds an $\omega_2$-Suslin tree. Related methods are Shelah's historic forcing and Todorcevic's $\rho$-functions.