Let T(Ω) be the ordinal notation system from Buchholz-Schütte (1988). [The order type of the countable segmentT(Ω)0 is — by Rathjen (1988) — the proof-theoretic ordinal the proof-theoretic ordinal ofACA 0 + (Π 1 l −TR).] In particular let ↦Ω a denote the enumeration function of the infinite cardinals and leta ↦ ψ0 a denote the partial collapsing operation on T(Ω) which maps ordinals of T(Ω) into the countable segment TΩ 0 of T(Ω). Assume that the (fast growing) extended Grzegorczyk (...) hierarchy $(F_a )_{a \in T(\Omega )_0 }$ and the slow growing hierarchy $(G_a )_{a \in T(\Omega )_0 }$ are defined with respect to the natural system of distinguished fundamental sequences of Buchholz and Schütte (1988) in the following way: $$\begin{array}{*{20}c} {G_0 (n): = 0,} & {F_0 (n): = (n + 1)^2 ,} \\ {\begin{array}{*{20}c} {G_{a + 1} (n): = G_a (n) + 1,} \\ {G_l (n): = G_{l[n]} (n),} \\ \end{array} } & {\begin{array}{*{20}c} {F_{a + 1} (n): = \underbrace {F_a (...F_a }_{n + 1 - times}(n)...),} \\ {F_l (n): = F_{l[n]} (n),} \\ \end{array} } \\ \end{array}$$ wherel is a countable limit ordinal (term) and (l[n]) n ∈N is the distinguished fundamental sequence assigned tol. For each ordinal (term)a in T(Ω) and each natural numbern letC n (a) be the formal term which results from the ordinal terma by successively replacing every occurence ofψ a by $\psi _{ - 1 + C_n (a)}$ whereψ −1 is considered as a defined function symbol, namely $\psi _{ - 1} b: = F_{\psi _0 b + 1} (n + 1)$ . (Note thatψ a 0=Ω a ) In this article it is shown that for each ordinal termψ 0 a in T(Ω) there exists a natural numbern 0 such thatψ 0 C n (a) ∈ T(Ω) and $G_{\psi _0 a} (n) \leqslant F_{\psi _0 C_n (a) + 1} (n + 1)$ holds for alln≥n 0. This hierarchy comparison theorem yields a plethora of new results on nontrivial lower bounds for the slow growing ordinals — i.e. ordinals for which the slow growing hierarchy yields a classification of the provably total functions of the theory in question — of various theories of iterated inductive definitions (and subsystems ofKPi) and on the number and size of the subrecursively inaccessible ordinals — i.e. ordinals at which the extended Grzegorczyk hierarchy and the slow growing hierarchy catch up — below the proof-theoretic ordinal ofACA 0+(Π 1 l −TR). In particular these subrecursively inaccessibles ordinals are necessarily of the form $\psi _0 \Omega ..._{\Omega _\omega }$. (shrink)

We give a self-contained and streamlined version of the classification of the provably computable functions of PA. The emphasis is put on illuminating as well as seems possible the intrinsic computational character of the standard cut elimination process. The article is intended to be suitable for teaching purposes and just requires basic familiarity with PA and the ordinals below ε0. (Familiarity with a cut elimination theorem for a Gentzen or Tait calculus is helpful but not presupposed).

We estimate the derivation lengths of functionals in Gödel's system \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $T$\end{document} of primitive recursive functionals of finite type by a purely recursion-theoretic analysis of Schütte's 1977 exposition of Howard's weak normalization proof for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $T$\end{document}. By using collapsing techniques from Pohlers' local predicativity approach to proof theory and based on the Buchholz-Cichon and Weiermann 1994 approach to subrecursive hierarchies we define a (...) collapsing f unction\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} ${\cal D}:T\to \omega$\end{document} so that for (closed) terms \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $a,b$\end{document} of Gödel's \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $T$\end{document} we have: If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $a$\end{document} reduces to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $b$\end{document} then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\omega>{\cal D}(a)>{\cal D}(b).$\end{document} By one uniform proof we obtain as corollaries: A derivation lengths classification for functionals in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $T$\end{document}, hence new proof of strongly uniform termination of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $T$\end{document}. A new proof of the Kreisel's classific ation of the number-theoretic functions which can be defined in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $T$\end{document}, hence a classification of the provably total functions of Peano Arithmetic. A new proof of Tait's results on weak normalization for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $T$\end{document}. A new proof of Troelstra's result on strong normalization for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $T$\end{document}. Additionally, a slow growing analysis of Gödel's \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $T$\end{document} is obtained via Girard's hierarchy comparison theorem. This analyis yields a contribution to two open pro blems posed by Girard in part two of his book on proof theory. For the sake of completeness we also mention the Howard Schütte bound on derivation lengths for the simple typed \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\lambda$\end{document}-calculus. (shrink)

The class of all recursive functions fails to possess a natural hierarchical structure, generated predicatively from "within". On the other hand, many (proof-theoretically significant) sub-recursive classes do. This paper attempts to measure the limit of predicative generation in this context, by classifying and characterizing those (predictably terminating) recursive functions which can be successively defined according to an autonomy condition of the form: allow recursions only over well-orderings which have already been "coded" at previous levels. The question is: how can a (...) recursion code a well-ordering? The answer lies in Girard's theory of dilators, but is reworked here in a quite different and simplified framework specific to our purpose. The "accessible" recursive functions thus generated turn out to be those provably recursive in $(\prod_{1}^{1}-CA)_{0}$. (shrink)

Apologies. The purpose of the following talk is to give an overview of the present state of aims, methods and results in Pure Proof Theory. Shortage of time forces me to concentrate on my very personal views. This entails that I will emphasize the work which I know best, i.e., work that has been done in the triangle Stanford, Munich and Münster. I am of course well aware that there are as important results coming from outside this triangle and I (...) apologize for not displaying these results as well.Moreover the audience should be aware that in some points I have to oversimplify matters. Those who complain about that are invited to consult the original papers.1.1. General. Proof theory startedwithHilbert's Programme which aimed at a finitistic consistency proof for mathematics.By Gödel's Theorems, however, we know that we can neither formalize all mathematics nor even prove the consistency of formalized fragments by finitistic means. Inspite of this fact I want to give some reasons why I consider proof theory in the style of Gentzen's work still as an important and exciting field of Mathematical Logic. I will not go into applications of Gentzen's cut-elimination technique to computer science problems—this may be considered as applied proof theory—but want to concentrate on metamathematical problems and results. In this sense I am talking about Pure Proof Theory.Mathematicians are interested in structures. There is only one way to find the theorems of a structure. Start with an axiom system for the structure and deduce the theorems logically. These axiom systems are the objects of proof-theoretical research. Studying axiom systems there is a series of more or less obvious questions. (shrink)

It is well known that the Ackermann function can be defined via diagonalization from an iteration hierarchy which is built on a start function like the successor function. In this paper we study for a given start function g iteration hierarchies with a sub-linear modulus h of iteration. In terms of g and h we classify the phase transition for the resulting diagonal function from being primitive recursive to being Ackermannian.