User blog comment:Edwin Shade/A Complete Analysis of Taranovsky's Notation/@comment-30118230-20180201150715/@comment-1605058-20180203130116

n-th order arithmetic is basically like the second-order arithmetic, except we also have variables for sets of sets of numbers etc. Zn is the formal theory of n-th order arithmetic, which I explain in detail here.

As for examples of proof-theoretic ordinal derivations, I'm afraid your two requests, it being in-depth and accessible, can't be simultaneously satisfied. Even the Peano arithmetic, the simplest nontrivial theory to analyze, is quite challenging (you can look at Pohler's notes to see how it looks like). Proof theory is, unfortunately, a field which is very demanding from a technical point of view.

For set theory, I know of a book Naive Set Theory by Halmos, but I haven't read it myself, so I can't really recommend it, but you might want to give it a look.