# 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

- 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