Year: 2026
Type: preprint
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 show that there is consistently a forcing destructible universally null set. 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 $\operatorname{add}(\mathcal{UN})<\operatorname{cov}(\mathcal{UN}) <\operatorname{non}(\mathcal{UN})<\operatorname{cof}(\mathcal{UN})$.
Source: arXiv (Cornell University)
Author: Goto, Tatsuya
Institution: TU Wien
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7168426753