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Axiom schema of replacement
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{{short description|Concept in set theory}} {{more footnotes|date=March 2013}} In [[set theory]], the '''axiom schema of replacement''' is a [[Axiom schema|schema]] of [[axiom]]s in [[Zermelo–Fraenkel set theory]] (ZF) that asserts that the [[image (mathematics)|image]] of any [[Set (mathematics)|set]] under any definable [[functional predicate|mapping]] is also a set. It is necessary for the construction of certain infinite sets in ZF. The axiom schema is motivated by the idea that whether a [[class (set theory)|class]] is a set depends only on the [[cardinality]] of the class, not on the [[rank (set theory)|rank]] of its elements. Thus, if one class is "small enough" to be a set, and there is a [[surjection]] from that class to a second class, the axiom states that the second class is also a set. However, because [[ZFC]] only speaks of sets, not proper classes, the schema is stated only for definable surjections, which are identified with their defining [[Well-formed formula|formulas]].
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