One of the most important principles of J.-Y. Girard's Π 2 1 -logic is induction on dilators. In particular, Girard used this principle to construct his famous functor Λ. He claimed that the totality of Λ is equivalent to the set existence axiom of Π 1 1 -comprehension from reverse mathematics. While Girard provided a plausible description of a proof around 1980, it seems that the very technical details have not been worked out to this day. A few years ago, a loosely related approach led to an equivalence between Π 1 1 -comprehension and a certain Bachmann-Howard principle. The present paper closes the circle. We relate the Bachmann-Howard principle to induction on dilators. This allows us to show that Π 1 1 -comprehension is equivalent to the totality of a functor J due to P. Päppinghaus, which can be seen as a streamlined version of Λ.

Abstract The notion of countable well order admits an alternative definition in terms of embeddings between initial segments. We use the framework of reverse mathematics to investigate the logical strength of this definition and its connection with Fraïssé’s conjecture, which has been proved by Laver. We also fill a small gap in Shore’s proof that Fraïssé’s conjecture implies arithmetic transfinite recursion over $\mathbf {RCA}_0$ , by giving a new proof of $\Sigma ^0_2$ -induction.

Abstract Transformations of well partial orders induce functions on the ordinals, via the notion of maximal order type. In most examples from the literature, these functions are not normal, in marked contrast with the central role that normal functions play in ordinal analysis and related work from computability theory. The present paper aims to explain this phenomenon. In order to do so, we investigate a rich class of order transformations that are known as $\textsf {WPO}$-dilators. According to a first main result of this paper, $\textsf {WPO}$-dilators induce normal functions when they satisfy a rather restrictive condition, which we call strong normality. Moreover, the reverse implication holds as well, for reasonably well-behaved $\textsf {WPO}$-dilators. Strong normality also allows us to explain another phenomenon: by previous work of Freund, Rathjen and Weiermann, a uniform Kruskal theorem for $\textsf {WPO}$-dilators is as strong as $\varPi ^1_1$-comprehension, while the corresponding result for normal dilators on linear orders is equivalent to the much weaker principle of $\varPi ^1_1$-induction. As our second main result, we show that $\varPi ^1_1$-induction is equivalent to the uniform Kruskal theorem for $\textsf {WPO}$-dilators that are strongly normal.

Abstract 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.

Abstract 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.

Abstract In a recent paper by M. Rathjen and the present author it has been shown that the statement “every normal function has a derivative” is equivalent to $\Pi ^1_1$ -bar induction. The equivalence was proved over $\mathbf {ACA_0}$ , for a suitable representation of normal functions in terms of dilators. In the present paper, we show that the statement “every normal function has at least one fixed point” is equivalent to $\Pi ^1_1$ -induction along the natural numbers.

Abstract 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.