Forcing axioms and the complexity of non-stationary ideals

We study the influence of strong forcing axioms on the complexity of the non-stationary ideal on [Formula: see text] and its restrictions to certain cofinalities. Our main result shows that the strengthening [Formula: see text] of Martin’s Maximum does not decide whether the restriction of the non-stationary ideal on [Formula: see text] to sets of ordinals of countable cofinality is [Formula: see text] -definable by formulas with parameters in [Formula: see text] . The techniques developed in the proof of this result also allow us to prove analogous results for the full non-stationary ideal on [Formula: see text] and strong forcing axioms that are compatible with [Formula: see text] . Finally, we answer a question of S. Friedman, Wu and Zdomskyy by showing that the [Formula: see text] -definability of the non-stationary ideal on [Formula: see text] is compatible with arbitrary large values of the continuum function at [Formula: see text] .