I am currently studying well-ordered classes in the context of NBG set theory and I am trying to find a well-ordered class that is not proper. Here is the relevant definition of the terminology (assuming one is already familiar with well-ordered classes):
- Given a class $A$ that is linearly ordered by $\le$, a subclass $B$ is called a lower section of $A$ iff all elements in $B$ are less than each element of $A$ not in $B$. We call $B$ a proper lower section if, in addition, $B \neq A$.
- We call a well-ordered class $A$ proper iff every proper lower section of $A$ is not only a class but also a set.
Thus, I am essentially trying to find a well-ordered class that has a proper lower section which is also a proper class. Initially, I immediately thought of the class of all ordinals $On$ but that turned out to be wrong. Right now, I am trying to construct a well-ordered class that has $On$ as a proper lower section but I am clueless as to how to do it. Could someone give me some pointers or suggest a different route?
Consider the following ordering $\prec$ on $\mathrm Ord$:
$$\alpha\prec\beta\iff\begin{cases}\beta=0,\alpha>0& \text{or}\\0<\alpha<\beta\end{cases}$$
Then $0$ has a proper class of predecessors. It's not hard to see that this is indeed a well-ordered class.