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axiom of extensionality
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(Axiom)
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If $X$ and $Y$ have the same elements, then $X = Y$ .
The Axiom of Extensionality is one of the axioms of Zermelo-Fraenkel set theory. In symbols, it reads: $$ \forall u(u \in X \leftrightarrow u \in Y) \rightarrow X = Y. $$ Note that the converse, $$ X = Y \rightarrow \forall u(u \in X \leftrightarrow u \in Y) $$ is an axiom of the predicate calculus. Hence we have, $$ X = Y \leftrightarrow \forall u(u \in X \leftrightarrow u \in Y). $$ Therefore the Axiom of Extensionality expresses the most fundamental
notion of a set: a set is determined by its elements.
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"axiom of extensionality" is owned by Sabean.
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extensionality |
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Cross-references: Calculus, predicate, converse, Zermelo-Fraenkel set theory, axioms
There are 7 references to this entry.
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 9937 times total.
Classification:
| AMS MSC: | 03E30 (Mathematical logic and foundations :: Set theory :: Axiomatics of classical set theory and its fragments) |
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Pending Errata and Addenda
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