# Definition:Minimal Infinite Successor Set/Definition 3

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## Definition

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.

## Also see

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

## Sources

- 1971: Gaisi Takeuti and Wilson M. Zaring:
*Introduction to Axiomatic Set Theory*: $7.28$