The axiom asserts the existence of a set containing the empty set and closed under the successor operation :In Zermelo–Fraenkel set theory, this supplies a set from which the set of natural numbers is constructed.
A set is inductive if it contains the empty set and contains whenever it contains . The axiom of infinity asserts that an inductive set exists. The natural numbers form the least inductive set: intersect one inductive set with the class of objects belonging to every inductive set. This use of separation gives a set, and the defining class is closed under successor.
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The Axiom of Infinity is one of the axioms of set theory, particularly in the context of Zermelo-Fraenkel set theory (ZF), which is a foundational system for mathematics. The Axiom of Infinity asserts the existence of an infinite set. Specifically, the axiom states that there exists a set \( I \) such that: 1. The empty set \( \emptyset \) is a member of \( I \).