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axiom of extensionality

(Axiom)

If and have the same elements, then .

The Axiom of Extensionality is one of the axioms of Zermelo-Fraenkel set theory. In symbols, it reads:

$\displaystyle \forall u(u \in X \leftrightarrow u \in Y) \rightarrow X = Y.$

Note that the converse,

$\displaystyle X = Y \rightarrow \forall u(u \in X \leftrightarrow u \in Y)$

is an axiom of the predicate calculus. Hence we have,

$\displaystyle X = Y \leftrightarrow \forall u(u \in X \leftrightarrow u \in Y).$

Therefore the Axiom of Extensionality expresses the most fundamental notion of a set: a set is determined by its elements.

"axiom of extensionality" is owned by Sabean.

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Other names:

extensionality

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Cross-references: Calculus, predicate, converse, Zermelo-Fraenkel set theory, axioms
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This is version 2 of axiom of extensionality, born on 2003-06-24, modified 2003-06-24.
Object id is 4391, canonical name is AxiomOfExtensionality.
Accessed 8537 times total.

Classification:

03E30 (Mathematical logic and foundations :: Set theory :: Axiomatics of classical set theory and its fragments)

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