Well ordering principles for iterated $\Pi^1_1$-comprehension

TL;DR

This paper introduces ordinal collapsing principles inspired by proof theory that are equivalent to iterated $ ext{Pi}^1_1$-comprehension and admissible sets, extending previous non-iterated results and connecting reverse mathematics with set theory.

Contribution

It establishes the equivalence of new ordinal collapsing principles with iterated $ ext{Pi}^1_1$-comprehension and admissible sets, advancing the understanding of their set-theoretic and proof-theoretic connections.

Findings

Ordinal collapsing principles are equivalent to iterated $ ext{Pi}^1_1$-comprehension.

The work extends previous non-iterated results to the iterated case.

Connects reverse mathematics with set-theoretic principles.

Abstract

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 $\Pi^1_1$-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 Montalb\'an's "Open questions in reverse mathematics" (Bull. Symb. Log. 17(3)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.