Definition:Cardinal Number

Definition

Let $S$ be a set.

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

$\vert S \vert = \displaystyle \bigcap \left\{ x \in \operatorname{On} : x \sim S \right\}$

where $\operatorname{On}$ is the class of all ordinals.

Compare cardinality.

Also see

• The definition of cardinal, where cardinals are defined as equivalence classes of sets rather than as ordinal numbers.

Sources

• 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