Coanalytic ultrafilter bases

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.