Axiom of extensionality
Axiom defining set equality by identical members.
Wikipedia / Wikimedia Commons
The axiom of extensionality, also known as the axiom of extent, appears in many versions of axiomatic set theory, including Zermelo–Fraenkel set theory. It essentially defines what a set is: informally, two sets are equal precisely when they contain the same members.
The term "extensionality" comes from logic. An intensional definition gives the necessary and sufficient conditions for a term to apply to an object—for instance, "an even number is an integer divisible by 2." An extensional definition, by contrast, lists every object the term applies to—for example, "an even number is any of the integers 0, 2, 4, 6, 8..., -2, -4, -6, -8..." In logic, the extension of a predicate is the set of all things for which that predicate holds true.
This logical concept was introduced into set theory in 1893 by Gottlob Frege in his *Basic Laws of Arithmetic*. He formally used the idea of an extension: for a predicate *F*, its extension (in German, *Umfang*) is the set of all objects satisfying *F*. For instance, if *F(x)* is "x is even," then its extension is the set {..., -4, -2, 0, 2, 4, ...}. Frege defined his infamous Basic Law V as: the extension of *F* equals the extension of *G* if and only if for all *x*, *F(x)* is equivalent to *G(x)*. This law states that if two predicates have the same extension—meaning they are satisfied by the same objects—then they are logically equivalent. However, it was later discovered that this axiom leads to Russell's paradox.
The first clear statement of the modern axiom of extensionality appeared in 1908, in a paper by Ernst Zermelo on the well-ordering theorem. There, Zermelo presented the first axiomatic set theory, now called Zermelo set theory, which became the foundation for modern set theories. The term Zermelo used for "extensionality" was *Bestimmtheit*. The English word "extensionality" only became common in mathematical and logical texts during the 1920s and 1930s, especially with the formalization of logic and set theory by figures such as Alfred Tarski and John von Neumann.
In the formal language of Zermelo–Fraenkel set theory, the axiom is written as: ∀x∀y [∀z (z ∈ x ↔ z ∈ y) → x = y]. In words: if the sets *x* and *y* have the same members, then they are the same set. In pure set theory, every member of a set is itself a set, though this is not the case in set theories that include urelements. The axiom's usefulnes
- first_explicit_statement
- 1908 by Ernst Zermelo
- original_term_by_Zermelo
- Bestimmtheit
- logical_term_introduced_to_set_theory
- 1893 by Gottlob Frege
- English_term_extensionality_became_commo
- 1920s and 1930s
- associated_figures
- Gottlob Frege, Ernst Zermelo, Alfred Tarski, John von Neumann, W.V. Quine
Lore & Background
In Zermelo–Fraenkel set theory, the axiom reads: ∀x∀y [∀z (z ∈ x ↔ z ∈ y) → x = y]. In Quine's New Foundations (NF) set theory, the treatment of equality differs. In Quine's 1937 paper, equality was defined as ∀z (x ∈ z → y ∈ z), and the principle of extensionality was given as a separate postulate. In his 1951 Mathematical Logic, Quine defined equality as ∀z (z ∈ x ↔ z ∈ y), exactly equivalent to the antecedent of the ZF axiom, and introduced a substitutivity axiom.
Reader's Guide
The axiom of extensionality is a foundational principle in set theory, establishing that a set is uniquely determined by its members. This axiom is crucial for the development of Zermelo–Fraenkel set theory, ensuring that sets with identical elements are identical. Its historical development traces from Frege's attempt to formalize extensions, which led to Russell's paradox, to Zermelo's explicit axiomatization in 1908. The axiom's formulation varies across set theories; in Quine's NF, equality is defined in terms of membership, with extensionality either as a separate postulate or built into the definition. The axiom's converse follows from the substitution property of equality, though it is sometimes given as a biconditional. The term 'extensionality' itself, rooted in logical distinctions between intensional and extensional definitions, became standard in the 1920s and 1930s through the work of Tarski and von Neumann. The axiom remains a cornerstone of modern set theory, providing a clear criterion for set identity.
Did You Know?
- The axiom of extensionality is also called the axiom of extent.
- Gottlob Frege introduced the logical term 'extension' to set theory in 1893 in his Basic Laws of Arithmetic.
- Ernst Zermelo first explicitly stated the modern axiom of extensionality in 1908, using the term 'Bestimmtheit'.
- In Quine's New Foundations (1937), the principle of extensionality was given as the postulate P1: x ⊂ y → (y ⊂ x → x = y).
Frequently Asked Questions
Who is Axiom of extensionality?
It is the foundational rule in axiomatic set theory stating that two sets are identical if and only if they share exactly the same elements. Ernst Zermelo gave it its first explicit formulation in 1908, originally labelling it with the German word 'Bestimmtheit.'
What are Axiom of extensionality's powers and role?
It acts as the identity criterion for sets, ensuring a set is fully determined by its membership rather than by any name, description, or construction process attached to it. Without it, set theory would lack a clear rule for deciding when two collections are actually the same object.
How does Axiom of extensionality's story end?
It remains a permanent fixture in virtually every standard set-theoretic framework, including Zermelo–Fraenkel set theory, and continues to underpin modern foundations. The English label 'extensionality' only settled into common mathematical usage during the 1920s and 1930s, well after Gottlob Frege introduced the underlying logical distinction in 1893.
Why is Axiom of extensionality important?
It is what grants sets their objecthood in the first place, because without a rule tying a set's identity to its members, the entire edifice of set theory would lack a criterion for equality. Foundationalists such as Tarski, von Neumann, and Quine all built their work on top of this principle.
Where does the name 'extensionality' come from?
The term is borrowed from logic, where an extensional definition pins down a term by listing every object it applies to, in contrast to an intensional definition that spells out necessary and sufficient conditions. Frege transplanted that logical distinction into the set-theoretic setting in 1893, and the English word gradually entered mathematical parlance by the 1920s.
More in Set Theory And Foundations 1-20
Spotted an error? Know more?
This is a living reference — every entry is fact-audited, and reader corrections feed straight into our audit queue. Suggest an edit · See this site's audit record