Class of Ordinals $On$ not a Set I have a elementary question about the Ordinals $On$. I following thread the Class of Ordinals is a set? is proved that the class $On$ can't be a set. One argument is: If we assume that $On$ is a set, hen it is well-ordered and transitive, hence an ordinal and an element of itself, there $\\{On\\} \in On$. Why does it contradicts to the wellorder? Wellorder says that every non empty subset $S \subset On$ has a minimal element. But if we consider the subset $\\{On\\} \subset On$ then I would say that it has itself as the minimal element since it is a single element there. Similary, if $\alpha \in On$ is considered as subset $\\{\alpha\\} \subset On$, then $\alpha$ is obviously the minimal element in $\\{\alpha\\}$, right? So why this argument deosn't work for $\\{On\\}$? Where is the error in my reasoning?

> One argument is: If we assume that $On$ is a set, hen it is well-ordered and transitive, hence an ordinal and an element of itself, there $\\{On\\} \in On$. Why does it contradicts to the wellorder?

This doesn't contradict well-orderedness (as Hagen von Eitzen's answer seems to mistakenly claim); it contradicts well- _foundedness_. The well-foundedness axiom of ZFC states (among other things) that no set is an element of itself.

The reasoning in the last paragraph of your question (where you asked "where is the error in my reasoning?") seems to be correct.