Forcing Axioms and the Definabilty of the Nonstationary Ideal on $\omega_1$

TL;DR

This paper investigates the definability of the nonstationary ideal on  under various strong set-theoretic assumptions, showing it cannot be --definable in some models but can be in others.

Contribution

It demonstrates the consistency of --definability of the nonstationary ideal under certain axioms and models, extending understanding of its definability properties.

Findings

Under -Main and a Woodin cardinal, --definability fails.

Under Woodin's ()-axiom, --definability also fails.

There exist models with -MA where  is --definable.

Abstract

We show that under $\BMM$ and "there exists a Woodin cardinal$"$, the nonstationary ideal on $\omega_1$ can not be defined by a $\Sigma_1$ formula with parameter $A \subset \omega_1$. We show that the same conclusion holds under the assumption of Woodin's $(\ast)$-axiom. We further show that there are universes where $\BPFA$ holds and $\NS$ is $\Sigma_1(\omega_1)$-definable. Last we show that if the canonical inner model with one Woodin cardinal $M_1$ exists, there is a universe where $\NS$ is saturated, $\Sigma_1(\omega_1)$-definable and $\MA$ holds.