User blog comment:Clarrity/List of ordinals (up to omega^omega)/@comment-30118230-20180225131945/@comment-34820539-20180225201054

If your intent is to create a list which shows an intuitively obvious listing of the ordinals in which anyone with sufficient knowledge can pick off from any given point in the list, then I'm afraid to tell you it isn't possible to do that up to the ordinal \(\omega_1^{CK}\) - and I'll tell you why.

The ordinal \(\omega_1^{CK}\) is the supremum of computable ordinals, (or the set of ordinals \(O\) for which you can do arithmetic on any ordinal less than or equal to a given ordinal \(\beta\in O\) in a finite amount of steps on some hypothetical computer), as you probably already know. What may be surprising however, is the fact that while people have defined a fundamental sequence for \(\omega_1^{CK}\), they have not defined a complete system of fundamental sequences for every ordinal less than or equal to \(\omega_1^{CK}\), meaning that while it is theoretically possible to calculate \(\omega_1^{CK}[n]\), or the nth term in the fundamental sequence of the Church-Kleene ordinal, it is unknown in nearly all cases how to calculate \((\omega_1^{CK}[n])[n]\), (a complete system of fundamental sequences is impossible I think).

When it comes to fundamental sequences for ordinals beneath \(\omega_1^{CK}\) then, it's really turtles all the way down! (Unless someone devises a way to create a complete system of fundamental sequences, but it remains to be seen.) If you want to make each new limit ordinal "flow" into the next as a generalization of the last which is intuitively seen as a generalization, it can still be done, but only provided you establish a definition of \(\omega_1^{CK}[n]\), provide the first few values, and then say in words that the ordinal are according to this sequence. Otherwise it is impossible for people to tell what you're saying.

On the other hand, if your intent is also showing the beauty of ordinals and their continual ascension then you might be interested in drawing a picture like the one you provided only which extends past \(\omega^{\omega\). You wouldn't be able to reach nearly as high as \(\omega_1^{CK}\), but you would make an interesting art piece while showing ordinal beauty in a more tangible format to those who think more pictorially. Try brainstorming a way to represent ordinals in the form of pictures that no one's thought of before and I'm sure you'll think of something.

Anyways, good fortune with the list!