finite set

mathematics

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June 26, 2026

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A finite set in mathematics is a collection of distinct objects, called elements, that contains a limited number of members. This means you can count the elements in the set, and the counting will eventually end.1

For example, the set of days in a week is a finite set because there are only seven days. In contrast, an infinite set has an unlimited number of elements, meaning you could keep counting forever and never reach an end. The set of all whole numbers (1, 2, 3, ...) is an example of an infinite set.23

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Georg Cantor
Set theory
…agreed that a set, whether finite or infinite, is a collection of objects (e.g., the integers, {0, ±1, ±2,…}) that share a particular property while each object retains its own individuality. But when Cantor applied the device of the one-to-one correspondence (e.g., {a, b, c} to {1, 2, 3}) to…
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…agreed that a set, whether finite or infinite, is a collection of objects (e.g., the integers, {0, ±1, ±2,…}) that share a particular property while each object retains its own individuality. But when Cantor applied the device of the one-to-one correspondence (e.g., {a, b, c} to {1, 2, 3}) to…
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set theory
Fundamental set concepts
In principle, any finite set can be defined by an explicit list of its members, but specifying infinite sets requires a rule or pattern to indicate membership; for example, the ellipsis in {0, 1, 2, 3, 4, 5, 6, 7, …} indicates that the list of natural numbers…
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In principle, any finite set can be defined by an explicit list of its members, but specifying infinite sets requires a rule or pattern to indicate membership; for example, the ellipsis in {0, 1, 2, 3, 4, 5, 6, 7, …} indicates that the list of natural numbers…
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metalogic
Satisfaction of a theory by a structure: finite and infinite models
…a “cardinal number,” which—for a finite set—is simply the number at which one stops in counting its elements. For infinite sets, however, the elements must be matched from set to set instead of being counted, and the “sizes” of these sets must thus be designated by transfinite numbers. A rather…
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…a “cardinal number,” which—for a finite set—is simply the number at which one stops in counting its elements. For infinite sets, however, the elements must be matched from set to set instead of being counted, and the “sizes” of these sets must thus be designated by transfinite numbers. A rather…
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