---
title: Cardinality
description: Cardinality measures the size of sets, distinguishing between different sizes of infinity. Learners will understand how to compare sets using bijections, define cardinal arithmetic, and apply Cantor's diagonal argument to prove the uncountability of the real numbers.
category: mathematics
subcategory: set-theory
difficulty: beginner, intermediate, advanced
url: /subject/cardinality
---

# Cardinality

Cardinality measures the size of sets, distinguishing between different sizes of infinity. Learners will understand how to compare sets using bijections, define cardinal arithmetic, and apply Cantor's diagonal argument to prove the uncountability of the real numbers.

## Available Resources

2 Videos • 2 Courses • 3 Websites

## Websites

### 1. nLab: Cardinality

The nLab wiki entry on cardinality, defining cardinal numbers as isomorphism classes of sets and discussing them from category-theoretic and constructive perspectives. Useful for readers who already know basic set theory and want the structural view of size.

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

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

**Tags:** cardinal-numbers, set-theory, category-theory, foundations-of-mathematics

### 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

## Videos

### 1. Hilbert's Infinite Hotel Paradox

A short animated TED-Ed lesson by Jeff Dekofsky that walks through Hilbert's paradox of the Grand Hotel. Viewers see why a full infinite hotel can still accommodate new guests, building intuition for countable infinity and one-to-one correspondence.

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

**Link:** https://www.youtube.com/watch?v=Uj3_KqkI9Zo

**Tags:** hilberts-hotel, countable-infinity, bijection, set-theory

### 2. How To Count Past Infinity

A Vsauce video by Michael Stevens explaining ordinal and cardinal numbers, from counting with omega to the distinction between countable and uncountable sets. Viewers come away understanding why some infinities are larger than others and how transfinite counting works.

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

**Link:** https://www.youtube.com/watch?v=SrU9YDoXE88

**Tags:** ordinal-numbers, cardinal-numbers, uncountable-sets, infinity

## Courses

### 1. Set Theory

Learn set theory & explore cardinality with Dr. Trefor Bazett's course. Master the fundamentals through this accessible YouTube playlist!

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

**Link:** https://www.youtube.com/playlist?list=PLHXZ9OQGMqxersk8fUxiUMSIx0DBqsKZS

**Tags:** discrete-mathematics, set-theory, logic, proofs, cardinality

### 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

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