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 $$\omega _2$$ ω 2 and its restrictions to certain cofinalities. Our main result shows that the strengthening $$\mathrm{{MM}}^{++}$$ MM + + of Martin’s Maximum does not decide whether the restriction of the non-stationary ideal on $$\omega _2$$ ω 2 to sets of ordinals of countable cofinality is $$\Delta _1$$ Δ 1 -definable by formulas with parameters in $$\mathrm{{H}}(\omega _3)$$ H ( ω 3 ) . The techniques developed in the proof of this result also allow us to prove analogous results for the full non-stationary ideal on $$\omega _2$$ ω 2 and strong forcing axioms that are compatible with $$\mathrm{{CH}}$$ CH . Finally, we answer a question of S. Friedman, Wu and Zdomskyy by showing that the $$\Delta _1$$ Δ 1 -definability of the non-stationary ideal on $$\omega _2$$ ω 2 is compatible with arbitrary large values of the continuum function at $$\omega _2$$ ω 2 .