Forcing Axioms and the Definabilty of the Nonstationary Ideal on ${\omega }_1$

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.

22 pages

Cited by in corpus (2)

• The global ${\Sigma }_{n+1}^1$-Uniformization Property and $\mathsf{BPFA}$
• PFA and the definability of the nonstationary ideal