PFA and precipitousness of the nonstationary ideal

TL;DR

This paper demonstrates that the Proper Forcing Axiom (PFA) does not necessarily lead to the precipitousness of the nonstationary ideal on , using Neeman's forcing with side conditions.

Contribution

It provides a novel application of Neeman's forcing method to separate PFA from the precipitousness of the nonstationary ideal.

Findings

PFA does not imply precipitousness of the nonstationary ideal on 

Neeman's forcing with side conditions is effective in this separation

The result clarifies the relationship between forcing axioms and ideal properties

Abstract

We apply Neeman's method of forcing with side conditions to show that PFA does not imply the precipitousness of the nonstationary ideal on $\omega_1$.