---
title: Ordinals
description: Ordinal numbers extend the natural numbers to describe the order types of well-ordered sets. Learners will understand the difference between cardinality and ordinality, perform ordinal arithmetic, and use ordinals to structure transfinite induction and recursion proofs.
category: mathematics
subcategory: set-theory
difficulty: beginner, intermediate, advanced
url: /subject/ordinals
---

# Ordinals

Ordinal numbers extend the natural numbers to describe the order types of well-ordered sets. Learners will understand the difference between cardinality and ordinality, perform ordinal arithmetic, and use ordinals to structure transfinite induction and recursion proofs.

## Available Resources

2 Courses • 3 Websites

## Websites

### 1. nLab: Ordinal

Reference article from the nLab research wiki giving the definition of ordinal numbers as well-ordered sets, their arithmetic, and constructive and structural variants. Suited to readers who know basic set theory and want precise statements, alternative formulations, and literature references.

**Difficulty:** Beginner | **Price:** Free

**Link:** https://ncatlab.org/nlab/show/ordinal

**Tags:** ordinals, set-theory, well-orders, category-theory

### 2. Stanford Encyclopedia of Philosophy - Set Theory

Joan Bagaria's peer-reviewed Stanford Encyclopedia of Philosophy entry surveying modern set theory: the ZFC axioms, ordinals and cardinals, the continuum hypothesis, forcing and independence results, large cardinals, and determinacy. Readers gain a map of the field's central questions and results.

**Difficulty:** Beginner | **Price:** Free

**Link:** https://plato.stanford.edu/entries/set-theory/

**Tags:** set-theory, zfc-axioms, cardinals-and-ordinals, forcing, large-cardinals, philosophy-of-mathematics

### 3. Wolfram MathWorld

**Author:** Eric W. Weisstein

MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading.

**Difficulty:** Intermediate | **Language:** English | **Price:** Free

**Link:** https://mathworld.wolfram.com

**Tags:** mathematics-reference, encyclopedia, abstract-algebra, number-theory, geometry

## Courses

### 1. Set Theory

Learn ordinal numbers with William Hamilton's Set Theory course! Explore fundamental concepts and deepen your understanding.

**Difficulty:** Advanced | **Price:** Free

**Link:** https://www.youtube.com/playlist?list=PLqVrd2LbnivhSZfwNWebxCe7925lTbIkt

**Tags:** set-theory, ordinals, cardinals, mathematical-logic

### 2. Introduction to Mathematical Thinking

**Author:** Keith Devlin

Stanford course by Keith Devlin on how mathematicians reason, starting with the logic of language (and, or, not, implication, quantifiers) and moving to the structure of proofs, including contradiction and induction. Learners become able to read, construct and critique simple mathematical proofs.

**Difficulty:** Beginner | **Language:** English | **Duration:** Expect to require at least 10 hours of study per week to complete this course satisfactorily. | **Price:** Free

**Link:** https://www.coursera.org/learn/mathematical-thinking

**Tags:** mathematical-logic, proof-writing, quantifiers, mathematical-reasoning

---

*This content is part of Dantes.io - Your Treasure Map to Knowledge*

*Curated by humans at Dantes.io. Personal study use welcome; republishing this curation requires permission (team@dantes.io).*

View this page online: https://dantes.io/subject/ordinals