Set (mathematics)
A set is a collection of distinct mathematical objects.
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In mathematics, a set is a collection of distinct items, known as its elements or members, which are usually mathematical objects like numbers, symbols, points, lines, shapes, variables, functions, or even other sets. Mathematics generally avoids giving a precise definition of "set" or "collection," since that would require relying on something already defined. Instead, sets are treated as basic objects whose behavior is governed by axioms based on intuitive ideas about collections, and nearly every other mathematical object is then defined strictly in terms of sets. The field of set theory examines different axiom systems and what follows from them. Since the early 1900s, the most common system has been ZFC (Zermelo–Fraenkel set theory with the axiom of choice).
Before the late 1800s, sets were not studied as a separate topic and were often confused with sequences. Most mathematicians viewed infinity only as potential—the result of an endless process—and were hesitant to accept infinite sets. For instance, a line was seen as a location where a point could be placed, not as a set of points. The mathematical investigation of infinite sets began with Georg Cantor, leading to surprising results and paradoxes. For example, the number line contains an infinite number of elements that is strictly larger than the infinite number of natural numbers, and any line segment has the same number of elements as the entire line. Assuming a set of all sets exists leads to a contradiction, known as Russell's paradox, which triggered a foundational crisis in mathematics and various proposed solutions. One of these, Zermelo–Fraenkel set theory, became the generally accepted foundation for set theory and all of mathematics, even though much of mathematics does not require its full strength. Meanwhile, sets became widely used across all mathematics. Algebraic structures and mathematical spaces are typically defined using sets, and many older mathematical results were restated in set terms. For example, Euclid's theorem is often phrased as "the set of prime numbers is infinite." David Hilbert predicted this widespread use when he said, "No one will drive us from the paradise that Cantor created for us."
This article summarizes the common rules and properties of sets used in mathematics, without relying on a specific logical framework. For the branch of mathematics focused on set
- field
- Mathematics
- known_for
- Foundational object in mathematics; studied by set theory; ZFC (Zermelo–Fraenkel set theory with the axiom of choice) is the most commonly used axiom system since the first half of the 20th century.
- key_concept
- A set is a collection of different things; elements belong to a set, denoted by ∈; the empty set has no elements; a singleton has exactly one element.
Lore & Background
Before the end of the 19th century, sets were not studied specifically and were not clearly distinguished from sequences. Most mathematicians considered infinity as potential and were reluctant to consider infinite sets. The mathematical study of infinite sets began with Georg Cantor (1845–1918), which provided counterintuitive statements and paradoxes, such as the number line having an infinite number of elements strictly larger than the infinite number of natural numbers, and any line segment having the same number of elements as the whole line. Assuming the existence of a set of all sets led to Russell's paradox, contributing to the foundational crisis of mathematics.
Reader's Guide
Sets are fundamental to modern mathematics. Since the first half of the 20th century, ZFC (Zermelo–Fraenkel set theory with the axiom of choice) has been the most commonly used axiom system, though much of mathematics does not require its full power. Sets are widely used in all mathematics; algebraic structures and mathematical spaces are typically defined in terms of sets, and many older mathematical results are restated in terms of sets, such as Euclid's theorem often stated as 'the set of the prime numbers is infinite'. The wide use of sets was prophesied by David Hilbert. The axiom of extensionality states that two sets are equal if and only if they have the same elements. A set can be specified by listing its elements (roster notation) or by giving a property that characterizes its elements. The empty set is a finite set with 0 elements; a set is finite if there exists a natural number n such that the first n natural numbers can be put in bijection with its elements.
Did You Know?
- A set may also be called a collection or family, especially when its elements are themselves sets.
- The empty set is denoted ∅, ∅, or {}; the sets {∅} and ∅ are different because the former has one element (∅) and the latter has no elements.
- Roster notation was introduced by Ernst Zermelo in 1908.
- The natural numbers form an infinite set, commonly denoted ℕ.
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