# Definition:Minimal Infinite Successor Set

## Definition

### Definition 1

Let $S$ be an infinite successor set.

The minimal infinite successor set $\omega$ is the infinite successor set given by:

$\omega := \displaystyle \bigcap \set {S' \subseteq S: \text{$S'$is an infinite successor set} }$

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

### Definition 2

The minimal infinite successor set $\omega$ is defined as the set of all finite ordinals:

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

### Definition 3

The minimal infinite successor set $\omega$ is defined as:

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

where $K_I$ is the class of all non-limit ordinals and $\operatorname{On}$ is the class of all ordinal numbers.

## Equivalence of Definitions

As shown in Equivalence of Definitions of Minimal Infinite Successor Set, the definitions above are equivalent.

## Also see

• Results about the minimal infinite successor set can be found here.