# Definition:Minimally Inductive Set/Definition 3

## Definition

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The **minimally inductive 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.

## Nomenclature

The name **minimally inductive set** is borrowed from the concept of the **minimally inductive class** as introduced by Raymond M. Smullyan and Melvin Fitting in their *Set Theory and the Continuum Problem*.

Keith Devlin, in *The Joy of Sets: Fundamentals of Contemporary Set Theory*, refers to this object as **the first infinite ordinal**.

Paul Halmos raises the concept in his *Naive Set Theory*, but fails to pin a name to it.

The term **minimal infinite successor set** was invented by $\mathsf{Pr} \infty \mathsf{fWiki}$ in an attempt to provide a name consistent and compatible with Halmos's approach.

As such, it is not generally expected to be seen in this context outside $\mathsf{Pr} \infty \mathsf{fWiki}$.

Instances of this term have subsequently been replaced by **minimally inductive set**.

## Also see

- Results about
**the minimally inductive set**can be found**here**.

## Sources

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