User blog comment:Scorcher007/S - Large Countable Ordinal Notation. Chapter I, Up to KPm./@comment-31580368-20190912145823/@comment-31580368-20190913125328

> in the declaration of axioms, we are not allowed to use the resulting theory itself. Oh, then this makes things very complicated. It may be necessary to redefine in another way the use of this hierarchy of ordinals in countable googology.

> I am still not understanding what "n∈O"-term precisely means. While I'm confused and won't use Θ. I’ll try to explain what I meant by another example.

Let's say we try to create OCF based on admissble ordinal. Limit OCF based on 1st admissble ordinal > ω is BHO, limit of OCF based on 2nd admissble ordinal > ω is bigger, and so on. If we have n admissbles ordinals (n∈ω), then limit OCF will be Buchholz ordinal. But if we have limit of admissble ordinals, then we can use it as a diagonalizer and limit OCF will be Takeuti-Feferman-Buchholz ordinal. n∈(ω-th admissble) means we can use n admissble ordinals (collection of admissble ordinals < ω-th admissble). n<(ω-th admissble) means we can use limit of admissble ordinals.