Cox, SeanLücke, Philipp2025-01-142025-01-142022-09-010026-9255https://hdl.handle.net/2445/217445We 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 $\mathrm{H}\left(\omega_3\right)$. 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 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 $\omega_2$.40 p.application/pdfengcc by (c) Sean cox et al., 2022http://creativecommons.org/licenses/by/3.0/es/Teoria de conjuntsLògica matemàticaSet theoryMathematical logicinfo:eu-repo/semantics/article7515772025-01-14info:eu-repo/semantics/openAccess