Filters and Ultrafilters in Real Analysis

TL;DR

This paper explores the use of filters and ultrafilters to define limits in real analysis, simplifying the traditional epsilon-delta approach by reducing the quantifier complexity through non-standard analysis techniques.

Contribution

It introduces a novel characterization of limits using filters and non-standard numbers, achieving a simpler, quantifier-efficient framework in real analysis.

Findings

Limits characterized with a single quantifier using filters.

Construction of the real non-standard number field.

Simplification of limit definitions compared to traditional methods.

Abstract

We study free filters and their maximal extensions on the set of natural numbers. We characterize the limit of a sequence of real numbers in terms of the Frechet filter, which involves only one quantifier as opposed to the three non-commuting quantifiers in the usual definition. We construct the field of real non-standard numbers and study their properties. We characterize the limit of a sequence of real numbers in terms of non-standard numbers which only requires a single quantifier as well. We are trying to make the point that the involvement of filters and/or non-standard numbers leads to a reduction in the number of quantifiers and hence, simplification, compared to the more traditional epsilon, delta-definition of limits in real analysis.