# ordinal numbers

## Summary

The ordinal numbers are a way of measuring the size of well-ordered sets. They give a way of talking about different sizes of infinity. They are needed to construct the cardinal numbers, which measure the sizes of sets more generally.

## Context

This concept has the prerequisites:

- natural numbers as sets (We must treat natural numbers as sets to use them as ordinal numbers.)
- well orderings (The ordinal numbers are well ordered, and defined using transfinite recursion.)
- set operations (The ordinal numbers are constructed with set union.)

## Goals

- Define what it means for a set to be an ordinal

- Know that the ordinals can be divided into limit and successor ordinals

- Show that the subset relation is a well ordering on the ordinals

- Define what it means for two well-ordered sets to be isomorphic

- Show that every well-ordered set is isomorphic to a unique ordinal

- Show that for every ordinal, there is a larger ordinal

## Core resources (read/watch one of the following)

## -Free-

→ Notes on Set Theory (2013)

Lecture notes for a course on axiomatic set theory.

## -Paid-

→ A Course in Mathematical Logic

A graduate textbook in mathematical logic.

Location:
Section 10.2, "Ordinals," pages 468-477

→ Elements of Set Theory

An introductory textbook on axiomatic set theory.

Location:
Chapter 7, "Orderings and ordinals," subsections "Epsilon-images" through "Ordinal numbers," pages 182-194

## See also

-No Additional Notes-