Definition:Ordinal/Definition 4

Definition

There is believed to be a mistake here, possibly a typo. In particular: Superinductive class does not assume that $g$ is the successor operation. I saw a few references to "superinductive with successor". Might we create a separate definition for that for ease and consistency of reference? You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by reviewing it, and either correcting it or adding some explanatory material as to why you believe it is actually correct after all. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Mistake}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Let $\alpha$ be a set.

$\alpha$ is an ordinal if and only if:

$\alpha$ is an element of every superinductive class.

Notation

The class of all ordinals can be found denoted $\On$.

In order to indicate that a set $S$ is an ordinal, this notation is often seen:

$\Ord S$

whose meaning is:

$S$ is an ordinal.

Thus $\operatorname {Ord}$ can be used as a propositional function whose domain is the class of all sets.

A tradition has grown up in set theory to use lowercase Greek letters $\alpha, \ \beta, \ \gamma, \ldots$ as symbols to represent variables over ordinals.

It is also customary to denote the ordering relation between ordinals as $\le$ rather than $\subseteq$ or $\preceq$.

Also see

• Equivalence of Definitions of Ordinal

Sources

• 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: Definition $1.1$