How much can "(recursively) large ordinal axioms" prove? In "Collapsing functions based on recursively large ordinals: A well–ordering proof for KPM", Michael Rathjen shows that certain notations for the proof-theoretic ordinals of theories, which previously were defined using large cardinals, in fact require only the "recursively large" counterparts of said cardinals, which are countable ordinals.
One is then led to wonder about theories analogous to large cardinal set theories, but with "miniaturized" versions of the axioms: for example, starting with a standard subsystem of second-order arithmetic, or perhaps Kripke-Platek set theory, and then adding on the existence of a certain recursively large ordinal instead of the corresponding large cardinal. Presumably there is a fairly simple relationship between what the two theories can prove.
Is this known and understood? If not, has it at least been studied?
 A: This is not a full answer, but:
The Stanford Encyclopedia of Philosophy's article on constructive and intuitionistic set theory has a section on "large sets" which gives a decent overview of the relevant analogues to large cardinal axioms in this context. This does not address analogues in other forms of set theory, let alone in other foundations.
Wikipedia's article on constructive set theory mentions that "Many modern results trace back to Rathjen and his students." Useful keywords include "admissible", "inaccessible", and "Mahlo", in particular when used in conjunction with the names of particular constructive set theories.
There is also at least some literature using terminology like this in conjunction with second- and higher- order arithmetic, as well as type theory. There seem to be close connections with the program of reverse mathematics.
Finally, this later question of mine received a comment from Andrej Bauer about recursion principles, as well as an answer from Nik Weaver's which mentions further work by Rathjen et al.
There are references to some of this in Wikipedia's article on ordinal analysis. There is also some relevant information in its article on large countable ordinals, in the section "Beyond recursive ordinals".
