Predicative collapsing principles

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.

This is the accepted version of a paper published in The Journal of Symbolic Logic