User blog comment:Deedlit11/Ordinal Notations IV: Up to a weakly inaccessible cardinal/@comment-7484840-20130709082828/@comment-5529393-20130709090837

To be precise, it is Kripke-Platek set theory with "there exists a recursively inaccessible ordinal", where an ordinal is recursively inaccessible if it is admissible and the limit of admissible ordinals. (Note that admissible ordinals are the recursive analogue of cardinals, so recursively inaccessible is the recursive analogue of weakly inaccessible cardinals.)

See

http://en.wikipedia.org/wiki/Kripke%E2%80%93Platek_set_theory

The proof theoretic ordinal of KP is the Bachmann-Howard ordinal.