# well orderings

## Summary

A total ordering R on a set S is a well ordering if every subset of S has a smallest element. Well orderings are important because one can use a generalization of mathematical induction known as transfinite induction. The canonical example is the ordinal numbers.

## Context

This concept has the prerequisites:

- order relations (Well orderings are a subclass of order relations.)
- set operations
- natural numbers as sets (One must view natural numbers as sets to treat ordinary mathematical induction as a special case of transfinite induction.)

## Goals

- Know what it means for a relation to be a well ordering

- Know the transfinite induction principle and why it is justified

- Be able to define functions using transfinite recursion

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

## -Free-

→ Notes on Set Theory (2013)

Lecture notes for a course on axiomatic set theory.

## -Paid-

→ Elements of Set Theory

An introductory textbook on axiomatic set theory.

Location:
Chapter 7, "Orderings and ordinals," subsection "Well orderings," pages 172-178

## See also

-No Additional Notes-