Spectra of uniformity

TL;DR

This paper investigates the properties and limitations of uniform ultrafilters on ordinals in the absence of the axiom of choice, including the spectrum of successors of regular cardinals with such ultrafilters.

Contribution

It establishes an Easton-like theorem for the spectrum of successors of regular cardinals with uniform ultrafilters and shows this spectrum need not be closed.

Findings

Spectrum of successors of regular cardinals with uniform ultrafilters can be characterized by an Easton-like theorem.

The spectrum of such cardinals is not necessarily closed.

Results hold without assuming the axiom of choice.

Abstract

We study some limitations and possible occurrences of uniform ultrafilters on ordinals without the axiom of choice. We prove an Easton-like theorem about the possible spectrum of successors of regular cardinals which carry uniform ultrafilters; we also show that this spectrum is not necessarily closed.