On the Consistency Strength of MM($\omega_1$)

TL;DR

This paper demonstrates that the consistency strength of Martin's Maximum restricted to partial orders of size  can be derived solely from ZFC consistency, simplifying its foundational assumptions.

Contribution

It establishes that MM() does not require stronger assumptions than ZFC, clarifying its foundational strength.

Findings

MM() is consistent relative to ZFC

No additional large cardinal assumptions are needed for MM()

Simplifies understanding of the strength of MM()

Abstract

We prove that the consistency strength of Martin's Maximum restricted to partial orders of cardinality $\omega_1$ follows from the consistency of ZFC.