Definition:Class of All Ordinals
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Definition
The class of all ordinals is defined, obviously enough, as the class of all ordinals:
- $\On = \leftset {x: x}$ is an ordinal $\rightset {}$
Therefore, by this definition, $A \in \On$ if and only if $A$ is an ordinal.
Also known as
The class of all ordinals is often referred to as the ordinal class, but this can be misconstrued as an ordinal class, which misrepresents it.
Also see
- Results about the class of all ordinals can be found here.
Sources
- 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $7.11$
- 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $5$: Ordinal Numbers: $\S 1$ Ordinal numbers