Filters and Ultrafilters in Real Analysis
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.
Keywords
Cite
@article{arxiv.1212.5740,
title = {Filters and Ultrafilters in Real Analysis},
author = {Max Garcia},
journal= {arXiv preprint arXiv:1212.5740},
year = {2012}
}