On rapid idempotent ultrafilters

2013/04/14 by Peter Krautzberger, Krautzberger, Peter
Mathematics · #03E75 #05D10 #54D80 #Advanced Topology and Set Theory #FOS: Mathematics #Limits and Structures in Graph Theory #Logic (math.LO) #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.1304.3955

openalex publication_date 2013/04/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This short note contains the proofs of two small but somewhat surprising results about ultrafilters on ℕ: 1. strongly summable ultrafilters are rapid, 2. every rapid ultrafilter induces a closed left ideal of rapid ultrafilters. As a consequence, there will be rapid minimal idempotents in all models of set theory with rapid ultrafilters.

Related

• Rapid ultrafilters and summable ideals
• Idempotent ultrafilters without Zorn's Lemma
• Minimal idempotent ultrafilters and the Auslander-Ellis theorem
• Q-points, selective ultrafilters, and idempotents, with an application to choiceless set theory
• The relation of rapid ultrafilters and Q-points to van der Waerden ideal