- I'm getting quite curious about what they call 'systems of notation' for countable ordinals - which means more or less what it sounds like, but was made into a precise notion by Church and Kleene. I'm curious about them, but I know very little about the formal theory of them. There's a well-known system of notation that can describe all
the ordinals up to ε_0. It's called Cantor normal form:http://en.wikipedia.org/wiki/Ordinal_arithmetic#Cantor_normal_form
(I apologize if you know all this stuff - someone out there won't.) Indeed the Wikipedia article says Cantor normal form describes all
ordinals... but, heh-heh, this notation involves ordinals, and when we get to ε_0 the Cantor normal form for ε_0 needs to mention ε_0! But before that point it's unproblematic.
However, I can imagine other systems of notation - not necessarily in the technical sense, which I don't understand - which are 'patchy', in that they can describe some ordinal but not other ordinals less than that. Those would certainly count as 'obscure' ordinals in your sense.
I don't know if the Veblen functions only give a 'patchy' system of notation, or can be used to give a notation that handles all
ordinals below the Fefferman-Schütte ordinal.