On rapid idempotent ultrafilters

This short note contains the proofs of two small but somewhat surprising results about ultrafilters on N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {N}$$\end{document}: (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. The history of this result has been published as an experiment in mathematical writing on the author’s website (Krautzberger, One Day in Colorado or Strongly Summable Ultrafilters are Rapid, 2012) and (Krautzberger, Rapid Idempotent Ultrafilters, 2012) where you can can also find additional remarks by Blass and Hindman, offering a form of peer-review.