User blog comment:Mh314159/Natural number recursion - first 4 rule sets/@comment-35470197-20191019143654/@comment-39541634-20191024172934

I think the next step is overcoming your fear of ordinals.

As P-bot already said, you'll need them to progress much further. And they are much simpler than you think they are. At this stage I recommend an intuitive approach, rather than learning the precise definitions of ordinals in set theory. You don't really need the latter approach until much much later.

By "an intuitive approach", I mean that you begin with getting a general "feel" of how ordinals work (with our guidance, of-course). They aren't that different from ordinary numbers, really. In fact, for many purposes, you can simply think of ω as "a number so large that it is larger than any finite number". And everything up to ε₀ is nothing more than sums, products and exponentials with ω's and finite numbers.

There are a few confusing quirks along the way, which is why you'll need the guidance of an experienced googologist. But if you proceed carefully, step-by-step, then you can get quite far with this approach.