# Idempotent ultrafilters without Zorn's Lemma

## TL;DR

This paper introduces additive filters and provides a new proof for the existence of idempotent ultrafilters on natural numbers without Zorn's Lemma, relying only on the Ultrafilter Theorem for the continuum.

## Contribution

It presents a novel proof of idempotent ultrafilters that avoids Zorn's Lemma, expanding the foundational understanding of ultrafilter existence.

## Key findings

- Existence of idempotent ultrafilters proved without Zorn's Lemma
- Introduction of the concept of additive filters
- Proof relies solely on the Ultrafilter Theorem for the continuum

## Abstract

We introduce the notion of additive filter and present a new proof of the existence of idempotent ultrafilters on N without any use of Zorn's Lemma, and where one only assumes the Ultrafilter Theorem for the continuum.