User blog comment:Deedlit11/An ordinal version of Bowers' notation/@comment-25418284-20130313014604/@comment-5529393-20130313023938

It's an interesting property, but think that the notations that specialize in the ordinals handle things much nicer than when we try to shoehorn a large number notation to the ordinals. A major reason for that is that they take advantage of fixed points of functions, rather than being hampered by them.

We can plug ordinals into the notation and use the same definitions. The problem for this notation (and it's a pretty general problem for all notations) is that the ordinals "stabilize" past a certain point - in this case, they stabilze at the next epsilon-number larger than b. We can go farther if we base the notation on "down-arrow notation" rather than "up-arrow notation" - that takes us to at least Gamma_0. But it may be that the ordinals stabilize further down the line - I'll have to investigate that.