axiom of replacement

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The axiom added to Zermelo's set theory by A. A. Fraenkel (1891–1965), to produce the classical set theory known as ZF. Put in terms of second-order logic, the axiom states that any function whose domain is a set has a range which is also a set. That is, if the arguments of a function form a set, so do the values of the function. This formulation is second-order because it quantifies over functions; in first-order logic the axiom needs to be stated as an axiom schema. See also Zermelo-Fraenkel set theory.

Subjects: Philosophy.

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