We study the definability of ultrafilter bases on $$\omega $$ in sense descriptive set theory. As a main result we show that there is no coanalytic base for Ramsey ultrafilter, while L can construct $$\Pi ^1_1$$ P-point and Q-point bases. also existence $${\varvec{\Delta }}^1_{n+1}$$ equivalent to $${\varvec{\Pi }}^1_n$$ base, $$n \in \omega . Moreover introduce Borel version classical number m...