In previous work, the author has shown that $\Pi ^1_1$ -induction along $\mathbb N$ is equivalent to a suitable formalization of the statement that every normal function on the ordinals has a fixed point. More precisely, this was proved for a representation of normal functions in terms of Girard’s dilators, which are particularly uniform transformations of well orders. The present paper works on the next type level and considers uniform transformations of dilators, which are called 2-ptykes. We show that $\Pi ^1_2$ -induction along $\mathbb N$ is equivalent to the existence of fixed points for all 2-ptykes that satisfy a certain normality condition. Beyond this specific result, the paper paves the way for the analysis of further $\Pi ^1_4$ -statements in terms of well ordering principles.

Harvey Friedman’s gap condition on embeddings of finite labelled trees plays an important role in combinatorics (proof of the graph minor theorem) and mathematical logic (strong independence results). In the present paper we show that the gap condition can be reconstructed from a small number of well-motivated building blocks: It arises via iterated applications of a uniform Kruskal theorem.

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 $1 + \beta \cdot \left( {\beta + \alpha } \right)$ (ordinal arithmetic) admits an almost order preserving collapse into β. Arithmetical comprehension is equivalent to a statement of the same form, with $\beta \cdot \alpha$ at the place of $\beta \cdot \left( {\beta + \alpha } \right)$. We will also characterize the principles that any set is contained in a countable coded ω-model of arithmetical transfinite recursion and arithmetical comprehension, respectively.