User blog comment:MilkyWay90/Help with understanding Veblen array notation/@comment-30754445-20180811202716/@comment-30754445-20180821115243

@Fefjo

"Thanks for the very clear explantion of computation of OCFs, I still find it less elegant that actual computation requires so much extra work but I guess thats to be expected from a system this powerful."

Computation with BHO-level OCFs don't take much work at all. I could give you the Veblen equivalent of any Madore-Psi expression you give me (up to the LVO), in a matter of seconds. I could also give you the fundamental sequences of such an expression in a similar time. There's actually a simple algorithm for finding fundamental sequences in this notation which you can learn very quickly (the downside of taking this shortcut is that you'll have no idea what you're doing. You'll get the right answer every time, but you won't know the actual ordinal values of the expressions you've written down).

To me, Veblen functions are much more cumbersome to work with, because there are so many rules to remember. All those pesky "+1's" and the differences between successor and limit ordinals whose manipulation depends on their position... now that's confusing! Even below the SVO, it isn't easy to remember all the exceptions (case in point: look at the SVO ruleset I've given above).

Returning to BHO-level OCFs, the part that does take extra work is the learning process required to master the notation. This is something you only need to do once. And as you said, this is expected from a system this powerful.