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 ${\mathbf{\Delta }}_{n+1}^1$ ultrafilter is equivalent to that of a ${\mathbf{\Pi }}_n^1$ ultrafilter base, for $n\in \omega$. Moreover we introduce a Borel version of the classical ultrafilter number and make some observations.