User blog comment:Hyp cos/Suggestions about Size Classes of Numbers/@comment-3427444-20170701135427/@comment-11227630-20170701230557

Another alternative naming system is to use names of number or set theories all the time. Then "tetration level" and "up-arrow notation level" will become "EFA level" and "PRA level" respectively.

However, I cannot find theories with proof-theoretic ordinal between $$\omega^\omega$$ (such as PRA, with growth rate limit $$\omega$$) and $$\varepsilon_0$$ (such as Peano arithmetic). Then there will be a large naming blank from "chained arrow notation level" to "quadramensional array level".