Abstract
We study the definability of ultrafilter bases on \(\omega \) in the sense of descriptive set theory. As a main result we show that there is no coanalytic base for a Ramsey ultrafilter, while in L we can construct \(\Pi ^1_1\) P-point and Q-point bases. We also show that the existence of a \({\varvec{\Delta }}^1_{n+1}\) ultrafilter is equivalent to that of a \({\varvec{\Pi }}^1_n\) ultrafilter base, for \(n \in \omega \). Moreover we introduce a Borel version of the classical ultrafilter number and make some observations.
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Acknowledgements
The author would like to thank the Austrian Science Fund, FWF, for generous support through START-Project Y1012-N35. We also thank the anonymous referee for many helpful comments that improved the paper.
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Schilhan, J. Coanalytic ultrafilter bases. Arch. Math. Logic 61, 567–581 (2022). https://doi.org/10.1007/s00153-021-00801-7
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DOI: https://doi.org/10.1007/s00153-021-00801-7