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Definition: A set of real numbers is called an inductive set if it has the following two properties: a) The number 1 is in the set. b) For every x in the set, the number x+1 is also in the set.
Source: Mathematical Analysis, second edition by Tom M. Apostol
Example: The set of all real numbers is an inductive set, as well as the set of all positive real numbers. The set of all positive integers is the smallest inductive set.