# Definition:Cardinality/Infinite

## Definition

Let $S$ be an infinite set.

The **cardinality** $\card S$ of $S$ can be indicated as:

- $\card S = \infty$

However, it needs to be noted that this just means that the cardinality of $S$ cannot be assigned a number $n \in \N$.

It means that $\card S$ is *at least* $\aleph_0$ (aleph null).

## Also defined as

Some authors, working to a particular mathematical agenda, do not discuss the **cardinality of an infinite set**, and instead limit their definition of this concept to the **finite case**.

Some others gloss over the definition for the **cardinality of a finite set**, perhaps on the understanding that the definition is trivial, and instead raise the concept only in the **infinite case**.

## Also known as

Some authors prefer the term **order** instead of **cardinality**, particularly in the context of finite sets.

Georg Cantor used the term **power** and equated it with the term **cardinal number**, using the notation $\overline {\overline M}$ for the **cardinality** of $M$.

Some sources cut through all the complicated language and call it the **size**.

Some sources use $\map {\#} S$ (or a variant) to denote **set cardinality**. This notation has its advantages in certain contexts, and is used on occasion on this website.

Others use $\map C S$, but this is easy to confuse with other uses of the same or similar notation.

A clear but relatively verbose variant is $\Card \paren S$ or $\operatorname{card} \paren S$.

1968: A.N. Kolmogorov and S.V. Fomin: *Introductory Real Analysis* use $\map m A$ for the **power** of the set $A$.

Further notations are $\map n A$ and $\overline A$.

## Also see

- Definition:Aleph Number ($\aleph_0, \aleph_1, \ldots$)
- Definition:Beth Number ($\beth_0, \beth_1, \ldots$)

## Sources

- 1968: A.N. Kolmogorov and S.V. Fomin:
*Introductory Real Analysis*... (previous) ... (next): $\S 2.5$: The power of a set - 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): Chapter $1$: Mappings: $\S 15$ - 1978: John S. Rose:
*A Course on Group Theory*... (previous) ... (next): $0$: Some Conventions and some Basic Facts - 1996: H. Jerome Keisler and Joel Robbin:
*Mathematical Logic and Computability*... (previous) ... (next): Appendix $\text{A}.6$: Cardinality

- 2012: M. Ben-Ari:
*Mathematical Logic for Computer Science*(3rd ed.) ... (previous) ... (next): Appendix $\text{A}.5$: Definition $\text{A}.25$