  We study the influence of strong forcing axioms on the complexity of the
non-stationary ideal on $\omega_2$ and its restrictions to certain
cofinalities. Our main result shows that the strengthening $MM^{++}$ of
Martin's Maximum does not decide whether the restriction of the non-stationary
ideal on $\omega_2$ to sets of ordinals of countable cofinality is
$\Delta_1$-definable by formulas with parameters in $H(\omega_3)$. The
techniques developed in the proof of this result also allow us to prove
analogous results for the full non-stationary ideal on $\omega_2$ and strong
forcing axioms that are compatible with CH. Finally, we answer a question of S.
Friedman, Wu and Zdomskyyshow by showing that the $\Delta_1$-definability of
the non-stationary ideal on $\omega_2$ is compatible with arbitrary large
values of the continuum function at $\omega_2$.
