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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?

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