Some Notes on Finite Sets

TL;DR

This paper introduces a set-theoretic approach to finite sets based on Kuratowski-finiteness, avoiding natural numbers, aimed at educational use and accessible to undergraduate mathematics students.

Contribution

It provides a gentle, natural-language-oriented account of finite sets using Kuratowski-finiteness, suitable for student seminars and without reliance on natural numbers.

Findings

Provides a clear, alternative foundation for finite sets

Extends previous work with revised explanations

Suitable for educational purposes

Abstract

These notes aim to give a gentle account to one approach to the theory of finite sets without making use of the natural numbers. They were written to be used as the basis for a student seminar. There are no real prerequisites except for a certain familiarity with the kind of mathematics seen in the first couple of years of a university mathematics course. The definition of being finite employed in these notes is usually called Kuratowski-finiteness and it is essentially that employed by Whitehead and Russell in Principia Mathematica. This is a revised and extended version of a paper with the same title from November 2015