# defining the cardinals

## Summary

Intuitively, one would like to define cardinal numbers as equivalence classes of sets, but unfortunately, these equivalence classes are too large to be sets. Instead, the cardinal numbers can be defined from the ordinal numbers. This construction requires the Axiom of Choice.

## Context

This concept has the prerequisites:

- cardinality
- Russell's Paradox (One cannot simply define cardinal numbers as equivalence classes because of the need to avoid Russell's Paradox.)
- functions and relations as sets (One must view relations as sets to see why you can't define cardinal numbers as equivalence classes.)
- ordinal numbers (The cardinal numbers are constructed from the ordinal numbers.)
- Axiom of Choice (Identifying the cardinals with ordinals requires the Axiom of Choice.)

## Goals

- Show that the Well-Ordering Theorem is equivalent to the Axiom of Choice

- Show that the trichotomy of the dominance relation is equivalent to the Well-Ordering Theorem

- Construct the cardinal numbers in terms of the ordinals

- Know why one cannot simply define cardinals as equivalence classes of sets under equinumerosity

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

## -Paid-

→ Elements of Set Theory

An introductory textbook on axiomatic set theory.

- Chapter 6, "Cardinal numbers and the Axiom of Choice," subsection "Axiom of Choice," pages 151-158
- Chapter 7, "Orderings and ordinals," subsection "Debts paid," pages 195-199

→ A Course in Mathematical Logic

A graduate textbook in mathematical logic.

Location:
Section 10.4, "Cardinality and the Axiom of Choice," pages 487-491

## Supplemental resources (the following are optional, but you may find them useful)

## -Free-

→ Notes on Set Theory (2013)

Lecture notes for a course on axiomatic set theory.

- Section 6, "The Axiom of Choice," only the parts dealing with the "Well-Ordering Theorem," pages 25-28
- Section 7, "Cardinals," pages 30-34
- Section 8, "Cardinal arithmetic," pages 34-40

## See also

-No Additional Notes-