# Predicative collapsing principles

## TL;DR

This paper explores the equivalence between arithmetical transfinite recursion and a formal collapsing principle involving ordinal arithmetic, providing new characterizations of these foundational logical systems.

## Contribution

It introduces a novel formal collapsing principle that characterizes arithmetical transfinite recursion and comprehension within ordinal arithmetic frameworks.

## Key findings

- Equivalence of arithmetical transfinite recursion with a formal collapsing principle.
- Characterization of arithmetical comprehension via a similar ordinal collapse.
- Description of principles ensuring sets are in countable models of these systems.

## Abstract

We show that arithmetical transfinite recursion is equivalent to a suitable formalization of the following: For every ordinal $\alpha$ there exists an ordinal $\beta$ such that $1+\beta\cdot(\beta+\alpha)$ (ordinal arithmetic) admits an almost order preserving collapse into $\beta$. Arithmetical comprehension is equivalent to a statement of the same form, with $\beta\cdot\alpha$ at the place of $\beta\cdot(\beta+\alpha)$. We will also characterize the principles that any set is contained in a countable coded $\omega$-model of arithmetical transfinite recursion resp. arithmetical comprehension.