On $\omega_3$-chains in P($\omega_1$) mod finite

TL;DR

This paper constructs specific ccc forcing notions under certain set-theoretic assumptions to add long chains and families of almost disjoint functions in the power set of ω₁ mod finite, using a novel three-dimensional iteration.

Contribution

It introduces a new three-dimensional finite support iteration technique to produce ccc forcings adding ω₃-chains and almost disjoint functions under the existence of a simplified (ω₁,2)-morass.

Findings

Existence of ccc forcing adding ω₃-chains in P(ω₁) mod finite.

Existence of ccc forcing adding ω₃-many strongly almost disjoint functions.

Use of a non-linear, three-dimensional iteration method.

Abstract

We prove that if there exists a simplified $(\omega_1,2)$-morass, then there is a ccc forcing which adds an $\omega_3$-chain in P($\omega_1$) mod finite and a ccc forcing which adds a family of $\omega_3$-many strongly almost disjoint functions from $\omega_1$ to $\omega$. The idea is to use a finite support iteration of countable forcings which is not linear but three-dimensional.