# Definition:Inductive Set

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## Definition

Let $S$ be a set of sets.

Then $S$ is **inductive** if and only if:

\((1)\) | $:$ | $S$ contains the empty set: | \(\ds \quad \O \in S \) | ||||||

\((2)\) | $:$ | $S$ is closed under the successor mapping: | \(\ds \forall x:\) | \(\ds \paren {x \in S \implies x^+ \in S} \) | where $x^+$ is the successor of $x$ | ||||

That is, where $x^+ = x \cup \set x$ |

### Axiomatic Set Theory

The concept of an **inductive set** is axiomatised in the Axiom of Infinity in Zermelo-Fraenkel set theory:

- $\exists x: \paren {\paren {\exists y: y \in x \land \forall z: \neg \paren {z \in y} } \land \forall u: u \in x \implies \paren {u \cup \set u \in x} }$

## Also known as

Some sources refer to this concept as a **successor set**.

However, note that this term has already been used on this site.

## Examples

### Inductive Set as Subset of Real Numbers

A specific instance of such an inductive set is defined by some authors as follows:

Let $I$ be a subset of the real numbers $\R$.

Then $I$ is an **inductive set** if and only if:

- $1 \in I$

and

- $x \in I \implies \paren {x + 1} \in I$

## Also see

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

- Results about
**inductive sets**can be found**here**.

## Sources

- 1960: Paul R. Halmos:
*Naive Set Theory*... (previous) ... (next): $\S 11$: Numbers - 2002: Thomas Jech:
*Set Theory*(3rd ed.) ... (previous) ... (next): Chapter $1$: Infinity