Coanalytic Ultrafilter Bases

TL;DR

This paper investigates the definability of ultrafilter bases on natural numbers, showing limitations for coanalytic bases of Ramsey ultrafilters and constructing definable bases in certain models, with implications for descriptive set theory.

Contribution

It proves the non-existence of coanalytic bases for Ramsey ultrafilters and constructs definable bases in the constructible universe, connecting ultrafilter existence to descriptive set-theoretic complexity.

Findings

No coanalytic base exists for a Ramsey ultrafilter.

In the constructible universe, $ ext{Pi}^1_1$ P-point and Q-point bases can be constructed.

Existence of $ ext{Delta}^1_{n+1}$ ultrafilters is equivalent to $ ext{Pi}^1_n$ 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 $\mathbf\Delta^1_{n+1}$ 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.