Foundations of Axiomatic Set Theory (Part II): Toward Infinity

hfwang132@gmail.com (hfwang132) — Wed, 24 Feb 2021 22:21:02 +0800

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Foundations of Axiomatic Set Theory (Part I): The Empty Set and Russell’s Paradox## Constructing Natural Numbers (ZF3: Axiom of Pairing, ZF4: Axiom of Union)

In the previous article, we successfully defined a unique set called the empty set and resolved Russell’s paradox. Next, we will use the empty set to construct more sets. To do so, we need more constructive axioms.

Axiom of Pairing: \(\forall a \forall b \exists c \forall x((x\in c)\leftrightarrow (x=a )\vee (x=b))\)

Foundations of Axiomatic Set Theory (Part I): The Empty Set and Russell's Paradox

hfwang132@gmail.com (hfwang132) — Sat, 23 Jan 2021 17:41:35 +0800

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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.

Set Theory on Haifei's Home

Foundations of Axiomatic Set Theory (Part II): Toward Infinity

Link to the Previous Article

Foundations of Axiomatic Set Theory (Part I): The Empty Set and Russell's Paradox

Preview

Introduction