Definition:Cardinal Number

Definition

Let $S$ be a set.

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

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

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

Compare cardinality.

Also see

• 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: $\S 10.7$
• 2008: Paul Halmos and Steven Givant: Introduction to Boolean Algebras ... (previous) ... (next): Appendix $\text{A}$: Set Theory: Cardinal Numbers
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Entry: cardinal number