Definition:Class of All Ordinals

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

• Class of All Ordinals is Ordinal

• 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