the Class of Ordinals is a set?

Question

Why the class of ordinals On is not a set?

Please dont tell me it's well-ordered till you tell me what you mean by an ordering on something that you dont know in first place if it is a set or not.

Answer 1

Assume On is a set. Then it is well-ordered and transitive, hence an ordinal and an element of itself, contradicting its own wellorder.

Answer 2

We don't need to know that something is a set in order to order it.

Formally speaking if $A$ is a class we can define an order on $A$ to be another class $R$ such that $z\in R$ means that $z=\langle x,y\rangle$ and $x,y\in A$, and moreover $R$ satisfies the axioms of an ordered set (whichever you prefer them to be, irreflexive and transitive; or reflexive, antisymmetric and transitive).

Now we can say that $\in\upharpoonright\sf Ord\times Ord$ is a well-ordering of $\sf Ord$.

---

But if we assume that $\sf Ord$ is a set then we can show that it is in fact an ordinal itself, therefore $\sf Ord\in Ord$. However this means that $\sf Ord$ is not well-founded, because $\{\sf Ord\}$ is a non-empty subset without a minimal element!