An **inductive set** is a fundamental concept in set theory and mathematical logic, particularly in the context of the natural numbers. A set \( S \) is called an inductive set if it satisfies two specific conditions: 1. **Base Element**: The set contains the base element, usually the number 0 (or 1, depending on the definition of natural numbers you are using).
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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.