Forcing and Cardinal Invariants of Universally Null Sets

Abstract

We study universally null sets in forcing extensions and the cardinal invariants of their ideal $\mathcal{UN}$. We prove that the Miller model contains a nonmeager universally null set of cardinality $\aleph_2=\mathfrak{c}$, answering a question of Brendle and Larson. The proof uses preservation of universal nullity by countable support iterations of Miller forcing and shows that the set of Miller reals added during the iteration is universally null. We also prove that every universally null set in the $\mathbb{PT}_{f,g}$ model has cardinality at most $\aleph_1$, and give a consistent example of a universally null set that acquires a perfect subset after an $\omega_1$-preserving forcing. For the cardinal invariants, we show that $|X|<\operatorname{cof}(\mathcal{UN})$ for every $X\in\mathcal{UN}$, and that $\operatorname{add}(\mathcal{N})=\mathfrak{c}$ implies $\operatorname{cof}(\mathcal{UN})=\mathfrak{d}_{\mathfrak{c}}$. Finally, we prove the consistency of $\operatorname{add}(\mathcal{UN})<\operatorname{cov}(\mathcal{UN}) <\operatorname{non}(\mathcal{UN})<\operatorname{cof}(\mathcal{UN})$.