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Predicative collapsing principles

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