On ${\omega }_3$-chains in P(${\omega }_1$) mod finite

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.

There are some gaps in the proof which I am unable to fix