Forcing axioms and the complexity of non-stationary ideals.

Abstract

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 ${\mathrm{MM}}^{++}$ 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 $H\left({\omega }_3\right)$ . 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 $\mathrm{CH}$ . Finally, we answer a question of S. Friedman, Wu and Zdomskyy 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 ${\omega }_2$ .