# Definition:Countable Set/Countably Infinite/Definition 1

## Definition

Let $S$ be a set.

$S$ is **countably infinite** if and only if there exists a bijection:

- $f: S \to \N$

where $\N$ is the set of natural numbers.

That is, it is an infinite set of the form:

- $\set {s_0, s_1, \ldots, s_n, \ldots}$

where $n$ runs over all the natural numbers.

An infinite set is **countably infinite** if it is countable, and is uncountable otherwise.

## Cardinality

The cardinality of a countably infinite set is denoted by the symbol $\aleph_0$ (**aleph null**).

## Also defined as

Some sources define a countable set to be what is defined on $\mathsf{Pr} \infty \mathsf{fWiki}$ as a countably infinite set.

That is, they use **countable** to describe a set which has *exactly the same* cardinality as $\N$.

Thus under this criterion $X$ is said to be countable if and only if there exists a bijection from $X$ to $\N$, that is, if and only if $X$ is equivalent to $\N$.

However, as the very concept of the term **countable** implies that a set **can be counted**, which, plainly, a finite set can be, it is suggested that this interpretation may be counter-intuitive.

Hence, on $\mathsf{Pr} \infty \mathsf{fWiki}$, the term countable set will be taken in the sense as to include the concept of finite set, and countably infinite will mean a countable set which is specifically *not* finite.

## Also known as

When the terms **denumerable** and **enumerable** are encountered, they generally mean the same as countably infinite.

Some pedagogues (for example Vi Hart and James Grime) use the term **listable**, but this has yet to catch on.

## Also see

## Sources

- 1964: Steven A. Gaal:
*Point Set Topology*... (previous) ... (next): Introduction to Set Theory: $2$. Set Theoretical Equivalence and Denumerability - 1965: Murray R. Spiegel:
*Theory and Problems of Laplace Transforms*... (previous) ... (next): Chapter $1$: The Laplace Transform: Some Special Functions: $\text {IX}$. Null functions - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Exercise $17.11$ - 1968: A.N. Kolmogorov and S.V. Fomin:
*Introductory Real Analysis*... (previous) ... (next): $\S 2.2$: Countable sets - 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): Chapter $1$: Mappings: $\S 15$ - 1972: A.G. Howson:
*A Handbook of Terms used in Algebra and Analysis*... (previous) ... (next): $\S 4$: Number systems $\text{I}$: A set-theoretic approach - 1981: Murray R. Spiegel:
*Theory and Problems of Complex Variables*(SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Point Sets: $15.$ - 1996: H. Jerome Keisler and Joel Robbin:
*Mathematical Logic and Computability*... (previous) ... (next): $\S 1.7$: Tableaus - 1996: H. Jerome Keisler and Joel Robbin:
*Mathematical Logic and Computability*... (previous) ... (next): Appendix $\text{A}.6$: Cardinality - 2000: James R. Munkres:
*Topology*(2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 7$: Countable and Uncountable Sets - 2008: Paul Halmos and Steven Givant:
*Introduction to Boolean Algebras*... (previous) ... (next): Appendix $\text{A}$: Set Theory: Countable Sets - 2010: Raymond M. Smullyan and Melvin Fitting:
*Set Theory and the Continuum Problem*(revised ed.) ... (previous) ... (next): Chapter $1$: General Background: $\S 2$ Countable or uncountable? - 2012: M. Ben-Ari:
*Mathematical Logic for Computer Science*(3rd ed.) ... (previous) ... (next): Appendix $\text{A}.5$: Definition $\text{A}.25$