Finite and Infinite Sets¶

Definition¶

Consider a set $A\text{.}$ We say $A$ is finite if it has $n$ distinct elements for some natural number $n$ and we write $|A|=n\text{;}$ otherwise, we say $A$ is infinite.

Interpretation¶

This distinction separates sets whose elements fit into a completed finite list $a_1,\ldots,a_n$ from sets for which no such finite list exists. For a finite set, $|A|$ is the length of that list.

The list counts distinct elements only, so $|\{1,1,2\}|=2$$2$. The case $n=0$ is allowed, which makes the empty set finite. A set is infinite when every finite list misses some of its elements: any list $a_1,\ldots,a_n$ of natural numbers misses $a_1+\cdots+a_n+1$$1$, so $\mathbb{N}$ is infinite.

Infinite sets behave differently when sizes are compared. The even natural numbers form a proper subset of $\mathbb{N}$, yet $n\mapsto2n$ pairs every natural number with exactly one even number; a finite set can never be paired in this way with a proper subset of itself. This comparison is the enrichment topic in the note below.

A common slip is to read "infinite" as "very large". A set with $10^{100}$ elements is finite, however long its list.

Key Results¶

• A finite set with $n$ distinct elements has cardinality $n\text{.}$
• The empty set is finite and has cardinality $0\text{.}$
• Every finite set of real numbers is bounded.

Examples¶

• $\{1,\,2,\,3\}$ is finite.
• $\mathbb{N}$ is infinite.

Prerequisites¶

• Set
• Natural Number

Related Concepts¶

• Cardinality
• Countable and Uncountable Sets

Sources¶

• Math I: Calculus · Lecture 1

Aliases: finite set, infinite set Symbols: natural numbers

Last updated: 20 October 2026 03:02