An approach to basic set theory and logic

TL;DR

This paper introduces a simplified set of axioms for basic set theory, including a novel 'Law of Extremes' axiom, to facilitate quick proofs and aid learning without complex logical nuances.

Contribution

It proposes a new axiom, 'The Law of Extremes,' enabling straightforward derivation of fundamental set theory facts and logical tautologies, simplifying foundational proofs.

Findings

Quick derivation of set-theoretic identities

Effective tool for memory and proof reinforcement

Simplified approach for teaching basic set theory

Abstract

The purpose of this paper is to outline a simple set of axioms for basic set theory from which most fundamental facts can be derived. The key to the whole project is a new axiom of set theory which I dubbed "The Law of Extremes". It allows for quick proofs of basic set-theoretic identities and logical tautologies, so it is also a good tool to aid one's memory. I do not assume any exposure to euclidean geometry via axioms. Only an experience with transforming algebraic identities is required. The idea is to get students to do proofs right from the get-go. In particular, I avoid entangling students in nuances of logic early on. Basic facts of logic are derived from set theory, not the other way around.