# cardinality

## Summary

Cardinality is a way of measuring the size of a set. Two sets have the same cardinality if they are equinumerous, i.e. if there is a bijective mapping between them. A has a larger cardinality than B if there is an injective mapping from B to A. Cardinality gives a precise way to talk about different sizes of infinity.

## Context

This concept has the prerequisites:

- countable sets (Cardinality generalizes the notion of countable and uncoutable sets.)
- equivalence relations (Cardinality is defined in terms of equinumerosity, an equivalence relation.)
- order relations (One must show that the dominance relation is a total ordering.)
- set operations (Sets of different cardinalities are constructed using various set operations.)
- natural numbers as sets (Defining cardinality for finite sets requires viewing natural numbers as sets.)

## Goals

- Define the relations of equinumerosity and dominance, and show that these are equivalence and order relations, respectively

- Prove the Schroeder-Bernstein Theorem: that if two sets each dominate each other, then they are equinumerous.

- Show that no set is equinumerous with its power set.

- Be able to manipulate cardinal numbers algebraically

- Be aware of the Continuum Hypothesis

## 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," up through subsection "Ordering cardinal numbers," pages 128-150
- Chapter 6, subsections "Arithmetic of infinite cardinals" and "Continuum Hypothesis," pages 162-166

## See also

-No Additional Notes-