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