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
Based on Takeuti
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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.