I am aware of three major "hierarchies" of mathematical theories, but I don't know how to relate these hierarchies to one another. Here are the hierarchies I have in mind:
Consistency strength. My understanding is that here one considers (recursively-enumerable?) theories $T$ of arithmetic (or which interpret the language of first-order arithmetic) of sufficient strength to fix some scheme for the syntax of first-order languages. One (partially) orders these theories by saying that $T > T'$ if $T$ proves the consistency of $T'$ (when $T'$ is coded up according to the aforementioned syntactic scheme).
Reverse mathematics. My understanding is that here one considers theories $T$ of second-order arithmetic (or which interpret the language of second-order arithmetic) and (partially) orders them directly by their implications.
Proof-theoretic ordinal analysis. My understanding is that here one considers theories $T$ of arithmetic (or which interpret the language of first-order arithmetic) and orders them by their proof-theoretic ordinal, i.e. the supremum of all (countable) ordinals $\alpha$ such that there exists a relation $R \subseteq \mathbb N \times \mathbb N$ definable in $T$ such that $T$ proves that $R$ is a well-order (although the sense in which $T$ can even express that $R$ is a well-order if $T$ is first-order is something that I don't quite understand), and $R$ is (externally) isomorphic to $\alpha$.
- Where do the domains of applicability of these hierachies overlap?
For instance, it seems that the "strongest" theories (like ZFC+ large cardinals) are usually studied in terms of consistency strength as opposed to reverse math or proof theory. I have the impression that proof-theoretic ordinals are most commonly used for relatively weak theories, and that reverse math lies somewhere in the middle. But I'm not even sure where to look for the overlaps in these domains, partly because the sorts of theories considered in each hierarchy are slightly different.
- Where there domains overlap, how do these hierarchies relate?
In general, I imagine there are no direct implications saying that any of these partial orders refines any of the others (even where their domains of applicability coincide). But I imagine that there are some general tendencies -- a stronger theory in one hierarchy should probably typically be stronger in another as well.
- Should I really be thinking of these 3 hierarchies as "comparable" in the sense that they give some notion of "strength" of a theory? And are there other hierarchies I should have in mind in this regard as well?