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:

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

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:

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

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

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