Definition:Cardinal Number

Definition

Let $S$ be a set.

The cardinal number of $S$ is defined as follows:

$\ds \card S = \bigcap \set {x \in \On : x \sim S}$

where $\On$ is the class of all ordinals.

Informal Definition

A natural number considered as an indication of the size of a set is referred to as a cardinal number.

Also known as

The cardinal number of a set $S$ is also known as the power or potency of $S$.

Also see

• Compare Definition:Cardinality.

• Definition:Cardinal, where cardinal numbers are defined as equivalence classes of sets rather than as ordinal numbers.

Historical Note

The concept of a cardinal number was first introduced by Georg Cantor.

Sources

• 1939: E.G. Phillips: A Course of Analysis (2nd ed.) ... (previous) ... (next): Chapter $\text I$: Number: $1.1$ Introduction
• 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 10.7$
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): number: 3.
• 2008: Paul Halmos and Steven Givant: Introduction to Boolean Algebras ... (previous) ... (next): Appendix $\text A$: Set Theory: Cardinal Numbers
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): number: 3.
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): cardinal number