$\omega$-morasses, and a weak form of Martin’s axiom provable in $\textrm {ZFC}$

Abstract:

We prove, in $\text {ZFC}$, that simplified gap-$1$ morasses of height $\omega$ exist. By earlier work on the relationship between morasses and forcing it immediately follows that a certain Martin’s axiom-type forcing axiom is provable in $\text {ZFC}$. We show that this forcing axiom can be thought of as a weak form of ${\text {MA}}_{\omega _1}$ and give some applications.