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Forcing and cardinal invariants of universally null sets

  • Tatsuya Goto
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Abstract

We study universally null sets in forcing extensions and the cardinal invariants of their ideal 𝒰𝒩. We prove that the Miller model contains a nonmeager universally null set of cardinality β„΅2=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 ℙ𝕋f,g model has cardinality at most β„΅1, and show that there is consistently a forcing destructible universally null set. For the cardinal invariants, we show that |X|<cof(𝒰𝒩) for every Xβˆˆπ’°π’©, and that add(𝒩)=c implies cof(𝒰𝒩)=dc. Finally, we prove the consistency of add(𝒰𝒩)<cov(𝒰𝒩) <non(𝒰𝒩)<cof(𝒰𝒩).

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Cite

BibTeXRIS
BibTeX
@misc{goto2026forcing,
  title = {{Forcing and cardinal invariants of universally null sets}},
  author = {Tatsuya Goto},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2607.12936},
  eprint = {2607.12936},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.12936},
}
RIS
TY  - UNPB
TI  - Forcing and cardinal invariants of universally null sets
AU  - Tatsuya Goto
PY  - 2026
DA  - 2026/10/09
PB  - arXiv
DO  - 10.48550/arxiv.2607.12936
UR  - https://arxiv.org/abs/2607.12936
AB  - We study universally null sets in forcing extensions and the cardinal invariants of their ideal 𝒰𝒩. We prove that the Miller model contains a nonmeager universally null set of cardinality β„΅_2=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 ℙ𝕋_f,g model has cardinality at most β„΅_1, and show that there is consistently a forcing destructible universally null set. For the cardinal invariants, we show that |X|<cof(𝒰𝒩) for every Xβˆˆπ’°π’©, and that add(𝒩)=c implies cof(𝒰𝒩)=d_c. Finally, we prove the consistency of add(𝒰𝒩)<cov(𝒰𝒩) <non(𝒰𝒩)<cof(𝒰𝒩).
ER  -