  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.
