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?



What is the order type of $L$ with Godel's well ordering?
The first question can be answered by taking the idea of definable well-orderings which are not set-like. That is, we can consider the formula $\varphi(x,y)$ which states that $x,y$ are distinct ordinals and either $y=0$ or $x\neq 0$ and $x\in y$. It is not hard to see that $\varphi$ defines a well-order of order type $\sf Ord+1$ on the ordinals. This can go on, and we can ask (e.g. [What is $\omega\_1^{CK}(\mathsf{Ord})$?](https://mathoverflow.net/questions/116590/what-is-omega-1ck-mathsford)):



> 
> Given $M$ a transitive model of $\sf ZFC$, what is the least ordinal $\alpha$ such that $\alpha$ is not a first-order definable over $M$?
> 
> 
> 


To the second question, the answer is in fact the the following things are equivalent:


1. The axiom of global choice.
2. Every class can be well-ordered.
3. There exists a well-ordering of $V$.


The implications $(3)\implies(2)\implies(1)$ is trivial. The proof that $(1)\implies(3)$ is about the same as the proof that the existence of a choice function on $\mathcal P(X)\setminus\{\varnothing\}$ implies that $X$ can be well-ordered.


Finally, the well-ordering of $L$ is of order type $\sf Ord$. To see this, simply note that if $x\in L\_\alpha$ then $x$ does not appear before $L\_\alpha$ in the order. Therefore we have a cofinal class of sets, which have only set-many predecessors. This is enough to show that there is no point which has class-many predecessors so the only order type fitting this is $\sf Ord$ itself.


