# Definition:Minimally Inductive Set

## Definition

### Definition 1

Let $S$ be an inductive set.

The **minimally inductive set** $\omega$ is the inductive set given by:

- $\ds \omega := \bigcap \set {S' \subseteq S: S' \text{ is an inductive set} }$

that is, $\omega$ is the intersection of every inductive set which is a subset of $S$.

### Definition 2

The **minimally inductive set** $\omega$ is defined as the set of all finite ordinals:

- $\omega := \set {\alpha: \alpha \text{ is a finite ordinal} }$

### Definition 3

The **minimally inductive set** $\omega$ is defined as:

- $\omega := \set {x \in \On: \paren {x \cup \set x} \subseteq K_I}$

where:

- $K_I$ is the class of all non-limit ordinals
- $\On$ is the class of all ordinals.

## Nomenclature

The name **minimally inductive set** is borrowed from the concept of the **minimally inductive class** as introduced by Raymond M. Smullyan and Melvin Fitting in their *Set Theory and the Continuum Problem*.

Keith Devlin, in *The Joy of Sets: Fundamentals of Contemporary Set Theory*, refers to this object as **the first infinite ordinal**.

Paul Halmos raises the concept in his *Naive Set Theory*, but fails to pin a name to it.

The term **minimal infinite successor set** was invented by $\mathsf{Pr} \infty \mathsf{fWiki}$ in an attempt to provide a name consistent and compatible with Halmos's approach.

As such, it is not generally expected to be seen in this context outside $\mathsf{Pr} \infty \mathsf{fWiki}$.

Instances of this term have subsequently been replaced by **minimally inductive set**.

## Also see

- Minimally Inductive Set Exists, demonstrating from Zermelo-Fraenkel set theory (ZF) that $\omega$ exists.

- Results about
**the minimally inductive set**can be found**here**.