${\sf MM}^{++}$ implies $(*)$

TL;DR

This paper proves that Martin's Maximum++ implies Woodin's P_max axiom (*), resolving a long-standing question and connecting two major set-theoretic axioms that both imply a rich structure of real numbers.

Contribution

It establishes that Martin's Maximum++ logically implies the P_max axiom (*), unifying two key axioms in set theory and answering a question from the 1990s.

Findings

Martin's Maximum++ implies P_max axiom (*)

Both axioms imply the existence of ℵ₂ many real numbers

The result unifies two major set-theoretic frameworks

Abstract

We show that Martin's Maximum${}^{++}$ implies Woodin's ${\mathbb P}_{\rm max}$ axiom $(*)$. This answers a question from the 1990's and amalgamates two prominent axioms of set theory which were both known to imply that there are $\aleph_2$ many real numbers.