Definition:Inductive Set

Definition
Let $S$ be a set of sets.

Then $S$ is inductive :


 * $(1): \quad \O \in S$
 * $(2): \quad \forall x: \paren {x \in S \implies x^+ \in S}$

where $x^+$ is the successor of $x$:
 * $x^+ = x \cup \set x$

Inductive Set as Subset of Real Numbers
A specific instance of such an inductive set is defined by some authors as follows:

Also see

 * Definition:Inductive Class, of which this is an instance