We introduce ordinal collapsing principles that are inspired by proof theory but have
a set theoretic flavor. These principles are shown to be equivalent to iterated 1
1-
comprehension and the existence of admissible sets, over weak base theories. Our
work extends a previous result on the non-iterated case, which had been conjectured
in Montalbán’s “Open questions in reverse mathematics" (Bull Symb Log 17(3):431–
454, 2011). This previous result has already been applied to the reverse mathematics
of combinatorial and set theoretic principles. The present paper is a significant contribution to a general approach that connects these fields
