Rethinking set theory

TL;DR

This paper explores adopting simple, practical set-theoretic principles as axioms, proposing a new foundational perspective that is accessible and grounded in everyday mathematical practice.

Contribution

It introduces a straightforward, practical set of axioms based on common principles, offering a new foundational approach inspired by Lawvere.

Findings

A set of ten mundane, practical axioms for set theory

Demonstrates how these axioms can serve as a reliable foundation

Provides an accessible expository perspective for mathematicians

Abstract

Mathematicians manipulate sets with confidence almost every day, rarely making mistakes. Few of us, however, could accurately quote what are often referred to as "the" axioms of set theory. This suggests that we all carry around with us, perhaps subconsciously, a reliable body of operating principles for manipulating sets. What if we were to take some of those principles and adopt them as our axioms instead? The message of this article is that this can be done, in a simple, practical way (due to Lawvere). The resulting axioms are ten thoroughly mundane statements about sets. This is an expository article for a general mathematical readership.