Preview

After reading this article, you will understand the first three axioms of ZF axiomatic set theory—the Axiom of Existence, the Axiom of Extensionality, and the Axiom Schema of Separation—and use them to resolve Russell’s paradox and define the empty set.

Introduction

We often hear statements such as “set theory is the foundation of modern mathematics.” In the eyes of many people, the set theory learned in secondary school consists merely of sets and simple operations—how could that be foundational? To understand this, we must understand how set theory was axiomatized.

Preface

Whenever physicists perform operations such as differentiating under the integral sign or interchanging the order of integration, the mathematicians watching can no longer sit still:

“Does your improper integral/series converge uniformly?”

The physicist replies:

“What is uniform convergence? We have always done it this way.”

Or:

“Assume that this function has sufficiently nice properties.”

Or, even more outrageously:

“Assume that this function looks rather pretty.”

looks good

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