Martin's maximum revisited

TL;DR

This paper explores the connection between stationary tower forcing and forcing axioms, showing that MM^{++} influences the -theory of H_{} under stationary set preserving forcings, extending Woodin's absoluteness results.

Contribution

It establishes that MM^{++} determines the -theory of H_{} for stationary set preserving forcings, linking forcing axioms with Woodin's theory.

Findings

MM^{++} decides the -theory of H_{} for stationary set preserving forcings

Connects stationary tower forcing with forcing axioms

Extends Woodin's absoluteness results to H_{}

Abstract

We present several results relating the general theory of the stationary tower forcing developed by Woodin with forcing axioms. The main results is that the forcing axiom MM^{++} (also known as MM^{+\omega_1}) decides the \Pi_2-theory of H_{\omega_2} with respect to stationary set preserving forcings. We argue that this is a close to optimal generalization to H_{\omega_2} of Woodin's absoluteness results for L(R).