TITLE: Ordinals that are not sets
QUESTION [11 upvotes]: The class of all ordinal numbers $\mathbf{Ord}$, aside being a proper class, can be thought of an ordinal number (of course it contains all ordinal numbers that are sets, not itself). Then one could consider $\mathbf{Ord}+1$, $\mathbf{Ord}+\mathbf{Ord}$, $\mathbf{Ord} \cdot \mathbf{Ord}$ and so on. Does this extension of ordinals make sense/is interesting? Maybe it was described by someone? Could it go deeper - to create a "superclass" of all ordinals that are classes?

REPLY [6 votes]: According to Kanamori, Reinhardt worked on extending the ordinals beyond the height of the universe. Kanamori talks about this in The Higher Infinite, page 313, where he cites the following reference from his bibliography.
Reinhardt, William N. Remarks on reflection principles, large cardinals, and elementary embeddings, pages 189-205
Jech, Thomas J. (ed.), Axiomatic Set Theory. Proceedings of Symposia in Pure Mathematics vol. 13, part 2. Providence, American Mathematical Society 1974.