User blog comment:Nayuta Ito/faketest/e0/@comment-30754445-20180804174000/@comment-30754445-20180804203535

Of-course it's possible.

Bird Arrays pretty much does this up ψ(Ω_Ω).

BTW There's a wonderful set of videos on youtube that "count" with Bird Arrays by Douglas Shamlin Junior. Every video starts counting where the previous video ended.

He started with 0 on April 2017, and every week or so a new video comes out with more numbers. Today he published Video #57 which reaches a little over ψ(ψ2(Ω)) in the usual notation.

As for UNOCF being "more intuitive for a newbie": This is problematic, because it doesn't work. Surely you can see the problem, when newbies base their intuitive understanding of ordinals on an ill-defined system? Especially one that seems - on the face of it - to be far weaker than it claims to be (if it can be turned into a well-defined notation at all)

And keep in mind another point:

The quest for larger numbers is equivalent to the quest for "bigger" and more complex ideas. Large ordinals, by their very nature, do not have a "simple and intuitive representation". In fact, ordinals are precisely the way mathematicians measure the complexity of your notation.

So there are no shortcuts here. As a somewhat banal example: if child doesn't know how to count yet, he won't be able understand what ω means, because understanding "ω" requires the concept of counting.

Similarly, if a googology newbie wants to understand how ε0 works, he will first need to learn a bunch of other stuff: Either ordinal arithmetic, or an equivalent array notation like BEAF. And either way, the resulting complexity will be about the same.

Which brings me to my main point:

A googology newbie cannot even begin to comprehend the BHO immediately. Anybody who is telling you otherwise is grossly misleading you (probably not intentionally). It's the very nature of the beast.

Don't be too disheartened, though. You can get there (and far beyond) after gaining a few months of hand-on experience. It isn't that difficult, actually. It doesn't require some kind of genius mind or anything. It's just that the BHO is "too big" for a person to grasp all at once.

And yes, I realize that many people here treat the BHO as if it's no big deal. Since the symbol ψ has become so popular here, nobody thinks that the limit of a single ψ function is impressive enough for take notice. Besides, it is much more exciting to talk about all those strange "inaccessible cardinals" and "stable ordinals" and proof theory and what's not...

But make no mistake: The BHO itself is HUGE. Case in point: the video series I've mentioned doesn't reach it until video 54, which is something like 13 hours into the "count". I'm telling you this as a person who understands the BHO-level extremely well. You have no idea how silly the situation here looks, from the point of view of an actual veteran googologist who spent years mastering the craft.

And if you're really serious about learning about these levels of numbers, please stop trying to take shortcuts. This will only hinder your progress in the long run, because you'll just fall into common misconceptions that you'll have to "unlearn" later.