2024/04/03 by Tom Benhamou, Benhamou, Tom, Fanxin Wu +1 · 2 citations
Mathematics · #Advanced Topology and Set Theory #FOS: Mathematics #General Topology (math.GN) #Logic (math.LO)
paper · pdf · doi:10.48550/arxiv.2404.02379
openalex publication_date 2024/04/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We provide two types of guessing principles for ultrafilter (\diamondsuit-λ(U), \diamondsuitpλ(U)) on ω which form subclasses of Tukey-top ultrafilters, and construct such ultrafilters in ZFC. These constructions are essentially different from Isbell's construction \citeIsbell65 of Tukey-top ultrafilters. We prove using the Borel-Cantelli Lemma that full guessing is not possible and rule out several stronger guessing principles e.g. we prove that no Dodd-sound ultrafilters exist on ω. We then apply these guessing principles to force a q-point which is Tukey-top (answering a question from \citeBenhanou/Dobrinen23), and prove that the class of ultrafilters which satisfy ¬\diamondsuit-λ is closed under Fubini sum. Finally, we show that \diamondsuit-λ and \diamondsuitpλ can be separated.