A hierarchy (J D g ) D Dilator of ordinal functionsJ D g : On→On is introduced and studied. It is a hierarchy of iterations relative to some giveng:OnarOn, defined by primitive recursion on dilators. This hierarchy is related to a Bachmann hierarchy $\left( {\phi _\alpha ^g } \right)_{\alpha< \varepsilon _{\Omega {\mathbf{ }} + {\mathbf{ }}1} }$ , which is built on an iteration ofg ↑ Ω as initial function.This Bachmann hierarchy $\left( {\phi _\alpha ^g } \right)_{\alpha< \varepsilon _{\Omega {\mathbf{ (...) }} + {\mathbf{ }}1} }$ itself is shown to be explicitly definable by $$\phi _\alpha ^g (\eta ) = g^{\Omega ^\alpha \cdot (1 + \eta )} (0)$$ from a hierarchy of iterationsg α :Ω→Ω for weakly monotonicg: Ω→Ω withg(0)>0. Forg=λη. (1+η) ·ω org=λη. 1, in particular, one obtains as $\left( {\phi _\alpha ^g } \right)_{\alpha< \varepsilon _{\Omega {\mathbf{ }} + {\mathbf{ }}1} }$ Bachmann hierarchy.With every ordinalα≦ε Ω+1 a dilatorD α is associated.D α is “below” the dilator $$\left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{2} + Id} \right)_{(\omega )}$$ , which is defined as $$\mathop {\sup }\limits_{n< \omega } (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{2} + Id)_{(n)}$$ with (¯2+Id)(0):=1 and $$(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{2} + Id)_{(n + 1)} : = (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{2} + Id)^{(2 + Id)_{(n)} }$$ . For weakly monotonicg:OnarOn satisfyingg(0)>0 and $g(\Omega ) \subseteqq \Omega$ , and forη<Ω andα<ε Ω+1 it is proved that. (shrink)

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. (shrink)

In this paper we give a new and comparatively simple proof of the following theorem by Girard [1]:“If ∀x∈ ${\cal O}$ ∃y∈ ${\cal O}$ ψ(x,y) (where the relationψ is arithmetic and positive in Kleene's ${\cal O}$ ), then there exists a recursive DilatorD such that ∀α≧ω∀x∈ ${\cal O}$ <α∃y∈ ${\cal O}$ (...) modification of Girard's system in [1]) we construct in a uniform way for each ordinalα a derivation Tα of the formula ∀x ∈ ${\cal O}$ <α∃y∈ ${\cal O}$ ψ(x, y), and then defineD immediately from the family (Tα)α∈On. Especially we set D(α):=Kleene-Brouwer length of (Tα). (shrink)