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Instances of this term have subsequently been replaced by minimally inductive set.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

• Results about the minimally inductive set can be found here.