# Definition:Minimal Infinite Successor Set

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## 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:

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

where:

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

## Equivalence of Definitions

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

## Also see

- Existence of Minimal Infinite Successor Set, demonstrating from Zermelo-Fraenkel set theory (ZF) that $\omega$ exists.

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