In some sense $Ord$ is a "proper class" ordinal. Unfortunately the notion of a proper class ordinal is not a straight forward generalization of the notion of "set" ordinals because the proper classes cannot be members of each other. But there are some hopes to define the notion of a "proper class" ordinal using a different view to the usual meaning of a "set" ordinal. For example if we find a property $P$ which is well defined on both sets and proper classes, and then discover a "theorem" like this:

"A set $s$ is an ordinal (in the usual sense) iff $s$ has the property $P$"

Then we can give a "definition" for the notion of a "proper class" ordinal which is compatible with the usual notion of "set" ordinals as follows:

"A class $C$ called a class-ordinal iff $C$ has the property $P$"

As an inexact suggestion in this direction note that the relations $\in$ and $\subsetneq$ are same on set ordinals, i.e.

$\forall \alpha,\beta \in Ord~~~~~\alpha\in \beta \Longleftrightarrow \alpha \subsetneq \beta$

And the relation $\subsetneq$ in well defined between classes. Now the question is:

**Question (1):** Is there a known generalization of the notion of set ordinals to proper classes?

If the answer of the above question is positive, then:

**Question (2):** Is there a proper class order type for any proper class well ordering using reasonable assumptions like the Axiom of Global Choice?

If the answer of the above question is positive too, then:

**Question (3):** What is the order type of $L$ with Godel's well ordering?

`Ord`

or`\mathrm{Ord}`

or`\mathsf{Ord}`

or similarly`ON`

or`ORD`

. It's just important that the reader is clear on the meaning and that the notation is consistent. $\endgroup$ – Asaf Karagila♦ Oct 6 '13 at 18:151more comment