Lifting proof theory to the countable ordinals II: second-order indescribable cardinals

TL;DR

This paper establishes a proof-theoretic reduction of the existence of Pi^{1}_{N}-indescribable cardinals to iterations of collapsings and Mahlo operations, providing bounds on definable countable ordinals in ZF set theory.

Contribution

It advances proof theory by linking second-order indescribable cardinals with iterative collapsing and Mahlo operations, offering new bounds within ZF.

Findings

Proof-theoretic reduction of Pi^{1}_{N}-indescribable cardinals.

Description of bounds on definable countable ordinals.

Connection between large cardinals and proof-theoretic operations.

Abstract

We show that the existence of a Pi^{1}_{N}-indescribable cardinal over the Zermelo-Fraenkel's set theory ZF is proof-theoretically reducible to iterations of Mostowski collapsings and lower Mahlo operations. Furthermore we describe a proof-theoretic bound on definable countable ordinals whose existence is provable from the existence of second order indescribable cardinals over ZF.