Definition:Non-Limit Ordinal
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Definition
An ordinal $\lambda$ is a non-limit ordinal if and only if it is not a limit ordinal.
That is, either the zero ordinal or a successor ordinal.
Notation
The class of all non-limit ordinals can be denoted $K_I$, while the class of all limit ordinals can be denoted $K_{II}$.
Also defined as
Some sources also consider the zero ordinal a limit ordinal.
It's a matter of taste.
Sources
- 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 7.27$
- 1993: Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (2nd ed.) ... (previous) ... (next): $\S 1$: Naive Set Theory: $\S 1.7$: Well-Orderings and Ordinals
- 2008: Paul Halmos and Steven Givant: Introduction to Boolean Algebras ... (previous) ... (next): Appendix $\text{A}$: Set Theory: Natural and Ordinal Numbers