The volume includes twenty-five research papers presented as gifts to John L. Bell to celebrate his 60th birthday by colleagues, former students, friends and admirers. Like Bell’s own work, the contributions cross boundaries into several inter-related fields. The contributions are new work by highly respected figures, several of whom are among the key figures in their fields. Some examples: in foundations of maths and logic ; analytical philosophy, philosophy of science, philosophy of mathematics and decision theory and foundations of economics. (...) Most articles are contributions to current philosophical debates, but contributions also include some new mathematical results, important historical surveys, and a translation by Wilfrid Hodges of a key work of arabic logic. (shrink)

In the 1920s, David Hilbert proposed a research program with the aim of providing mathematics with a secure foundation. This was to be accomplished by first formalizing logic and mathematics in their entirety, and then showing---using only so-called finitistic principles---that these formalizations are free of contradictions. ;In the area of logic, the Hilbert school accomplished major advances both in introducing new systems of logic, and in developing central metalogical notions, such as completeness and decidability. The analysis of unpublished material presented (...) in Chapter 2 shows that a completeness proof for propositional logic was found by Hilbert and his assistant Paul Bernays already in 1917--18, and that Bernays's contribution was much greater than is commonly acknowledged. Aside from logic, the main technical contribution of Hilbert's Program are the development of formal mathematical theories and proof-theoretical investigations thereof, in particular, consistency proofs. In this respect Wilhelm Ackermann's 1924 dissertation is a milestone both in the development of the Program and in proof theory in general. Ackermann gives a consistency proof for a second-order version of primitive recursive arithmetic which, surprisingly, explicitly uses a finitistic version of transfinite induction up to www . He also gave a faulty consistency proof for a system of second-order arithmetic based on Hilbert's &egr;-substitution method. Detailed analyses of both proofs in Chapter 3 shed light on the development of finitism and proof theory in the 1920s as practiced in Hilbert's school. ;In a series of papers, Charles Parsons has attempted to map out a notion of mathematical intuition which he also brings to bear on Hilbert's finitism. According to him, mathematical intuition fails to be able to underwrite the kind of intuitive knowledge Hilbert thought was attainable by the finitist. It is argued in Chapter 4 that the extent of finitistic knowledge which intuition can provide is broader than Parsons supposes. According to another influential analysis of finitism due to W. W. Tait, finitist reasoning coincides with primitive recursive reasoning. The acceptance of non-primitive recursive methods in Ackermann's dissertation presented in Chapter 3, together with additional textual evidence presented in Chapter 4, shows that this identification is untenable as far as Hilbert's conception of finitism is concerned. Tait's conception, however, differs from Hilbert's in important respects, yet it is also open to criticisms leading to the conclusion that finitism encompasses more than just primitive recursive reasoning. (shrink)

We survey a selection of results about majorization hierarchies. The main focus is on classical and recent results about the comparison between the slow and fast growing hierarchies.

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 article investigates a system of polymorphically typed combinatory logic which is equivalent to Gödel’s T. A notion of (strong) reduction is defined over terms of this system and it is proved that the class of well-formed terms is closed under both bracket abstraction and reduction. The main new result is that the number of contractions needed to reduce a term to normal form is computed by an ε0-recursive function. The ordinal assignments used to obtain this result are also used (...) to prove that the system under consideration is indeed equivalent to Gödel’s T. It is hoped that the methods used here can be extended so as to obtain similar results for stronger systems of polymorphically typed combinatory terms. An interesting corollary of such results is that they yield ordinally informative proofs of normalizability for sub-systems of second-order intuitionist logic, in natural deduction style. (shrink)

Sections 1 through 4 define, in the usual inductive style, various classes of object including one which is called the “combinatory terms of polymorphic type”. Section 5 defines a reduction relation on these terms. Section 6 shows that the weak normalizability of the combinatory terms of polymorphic type entails the weak normalizability of the lambda terms of polymorphic type. The entailment is not vacuous, because the combinatory terms of polymorphic type are indeed weakly normalizable, as is proven in Sect. 7 (...) using Tait and Girard’s computability predicates. The remainder of the paper is devoted to arguing that interesting consequences would follow from the existence of an “ordinally informative” proof, i.e. one which uses transfinite induction over recursive ordinal numbers and otherwise finitary methods, of normalizability for as large a class of the terms as possible. (shrink)

This article investigates a theory of type assignment (assigning types to lambda terms) called ETA which is intermediate in strength between the simple theory of type assignment and strong polymorphic theories like Girard’s F (Proofs and types. Cambridge University Press, Cambridge, 1989). It is like the simple theory and unlike F in that the typability and type-checking problems are solvable with respect to ETA. This is proved in the article along with three other main results: (1) all primitive recursive functionals (...) of finite type are representable in ETA; (2) every term typable in ETA has a unique normal form; (3) there is a function defined by ${{\varepsilon}_0}$ -recursion which takes every typable term to a natural number which is an upper bound to the lengths of all βη-reduction sequences starting with that term. (shrink)

By taking a closer look at the construction of an Ackermann function we see that between any primitive recursive degree and its Ackermann modification there is a dense chain of primitive recursive degrees.

Herbrand's Theorem, in the form of $$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\exists } $$ -inversion lemmata for finitary and infinitary sequent calculi, is the crucial tool for the determination of the provably total function(al)s of a variety of theories. The theories are (second order extensions of) fragments of classical arithmetic; the classes of provably total functions include the elements of the Polynomial Hierarchy, the Grzegorczyk Hierarchy, and the extended Grzegorczyk Hierarchy $\mathfrak{E}^\alpha $ , α < ε0. A subsidiary aim of the paper is to show (...) that the proof theoretic methods used here are distinguished by technical elegance, conceptual clarity, and wide-ranging applicability. (shrink)

Hilbert developed his famous finitist point of view in several essays in the 1920s. In this paper, we discuss various extensions of it, with particular emphasis on those suggested by Hilbert and Bernays in Grundlagen der Mathematik (vol. I 1934, vol. II 1939). The paper is in three sections. The first deals with Hilbert's introduction of a restricted ? -rule in his 1931 paper ?Die Grundlegung der elementaren Zahlenlehre?. The main question we discuss here is whether the finitist (meta-)mathematician would (...) be entitled to accept this rule as a finitary rule of inference. In the second section, we assess the strength of finitist metamathematics in Hilbert and Bernays 1934. The third and final section is devoted to the second volume of Grundlagen der Mathematik. For preparatory reasons, we first discuss Gentzen's proposal of expanding the range of what can be admitted as finitary in his esssay ?Die Widerspruchsfreiheit der reinen Zahlentheorie? (1936). As to Hilbert and Bernays 1939, we end on a ?critical? note: however considerable the impact of this work may have been on subsequent developments in metamathematics, there can be no doubt that in it the ideals of Hilbert's original finitism have fallen victim to sheer proof-theoretic pragmatism. (shrink)

.A term rewrite system is called simply terminating if its termination can be shown by means of a simplification ordering. According to a result of Weiermann, the derivation length function of any simply terminating finite rewrite system is eventually dominated by a Hardy function of ordinal less than the small Veblen ordinal. This bound had appeared to be of rather theoretical nature, because all known examples had had multiple recursive complexities, until recently Touzet constructed simply terminating examples with complexities beyond (...) multiple recursion. This was established by simulating the Hydra battle for all ordinal segments below the proof-theoretic ordinal of Peano arithmetic. By extending this result to the small Veblen ordinal we prove the huge bound of Weiermann to be sharp. As a spin-off we can show that total termination allows for complexities as high as those of simple termination. (shrink)

Finitism is given an interpretation based on two ideas about strings (sequences of symbols): a replacement principle extracted from Hilberts class 2 can be justified by means of an additional finitistic choice principle, thus obtaining a second equational theory . It is unknown whether is strictly stronger than since 2 may coincide with the class of lower elementary functions.

It can be argued that only the equational theories of some sub-elementary function algebras are finitistic or intuitive according to a certain interpretation of Hilbert's conception of intuition. The purpose of this paper is to investigate the relation of those restricted forms of equational reasoning to classical quantifier logic in arithmetic. The conclusion reached is that Edward Nelson's ‘predicative arithmetic’ program, which makes essential use of classical quantifier logic, cannot be justified finitistically and thus requires a different philosophical foundation, possibly (...) as a restricted form of logicism. (shrink)

In his 1953's paper, Grzegorczyk proved that a certain kind of relation classes of Grzegorczyk's hierarchy could be characterized inductively. We give a simpler version of this characterization.

We define a class of functions, the descent recursive functions, relative to an arbitrary elementary recursive system of ordinal notations. By means of these functions, we provide a general technique for measuring the proof-theoretic strength of a variety of systems of first-order arithmetic. We characterize the provable well-orderings and provably recursive functions of these systems, and derive various conservation and equiconsistency results.

This paper presents a new method - which does not rely on the cut-elimination theorem - for characterizing the provably total functions of certain intuitionistic subsystems of arithmetic. The new method hinges on a realizability argument within an infinitary language. We illustrate the method for the intuitionistic counterpart of Buss's theory Smath image, and we briefly sketch it for the other levels of bounded arithmetic and for the theory IΣ1.

A generalization is given of McAloon's result on initial segments ofmodels of GlΔ0, the fragment of Peano Arithmetic where the induction scheme is restricted to formulas with bounded quantifiers.

A realizability notion that employs only Kalmar elementary functions is defined, and, relative to it, the soundness of EA-(Π₁⁰-IR), a fragment of Heyting Arithmetic (HA) with names and axioms for all elementary functions and induction rule restricted to Π₁⁰ formulae, is proved. As a corollary, it is proved that the provably recursive functions of EA-(Π₁⁰-IR) are precisely the elementary functions. Elementary realizability is proposed as a model of strict arithmetic constructivism, which allows only those constructive procedures for which the amount (...) of computational resources required can be bounded in advance. (shrink)

This paper investigates the status of the fragments of Peano Arithmetic obtained by restricting induction, collection and least number axiom schemes to formulas which are Δ1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta_1}$$\end{document} provably in an arithmetic theory T. In particular, we determine the provably total computable functions of this kind of theories. As an application, we obtain a reduction of the problem whether IΔ0+¬exp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${I\Delta_0 + \neg \mathit{exp}}$$\end{document} implies BΣ1\documentclass[12pt]{minimal} (...) \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${B\Sigma_1}$$\end{document} to a purely recursion-theoretic question. (shrink)

Let Iπ2 denote the fragment of Peano Arithmetic obtained by restricting the induction scheme to parameter free Π2Π2 formulas. Answering a question of R. Kaye, L. Beklemishev showed that the provably total computable functions of Iπ2 are, precisely, the primitive recursive ones. In this work we give a new proof of this fact through an analysis of certain local variants of induction principles closely related to Iπ2. In this way, we obtain a more direct answer to Kaye's question, avoiding the (...) metamathematical machinery needed for Beklemishev's original proof.Our methods are model-theoretic and allow for a general study of Iπn+1 for all n≥0n≥0. In particular, we derive a new conservation result for these theories, namely that Iπn+1 is πn+2 -conservative over IΣn for each n≥1n≥1. (shrink)

We define theories of bounded arithmetic, whose definable functions and relations are exactly those in certain complexity classes. Based on a recursion-theoretic characterization of NC in Clote , the first-order theory TNC, whose principal axiom scheme is a form of short induction on notation for nondeterministic polynomial-time computable relations, has the property that those functions having nondeterministic polynomial-time graph Θ such that TNC x y Θ are exactly the functions in NC, computable on a parallel random-access machine in polylogarithmic parallel (...) time with a polynomial number of processors.0 We then define three theories of weak second-order arithmetic which respectively characterize relations in the classes of alternating logarithmic time, logspace and nondeterministic logspace. (shrink)

Recently, G. ASSER has obtained two interesting characterizations of the class of unary primitive recursive string-functions over a fixed alphabet as Robinson algebras. Both characterizations use a somewhat artificial string-function, namely the string-function lexicographically associated with the number-theoretical excess-over-a-square function. Our aim is to offer two new and natural Robinson algebras which are equivalent to ASSER’S algebras.

In this article we give a unifying approach to the theory of fundamental sequences and their related Hardy hierarchies of number-theoretic functions and we show the equivalence of the new approach with the classical one.

Let T be Gödel's system of primitive recursive functionals of finite type in a combinatory logic formulation. Let $T^{\star}$ be the subsystem of T in which the iterator and recursor constants are permitted only when immediately applied to type 0 arguments. By a Howard-Schütte-style argument the $T^{\star}$ -derivation lengths are classified in terms of an iterated exponential function. As a consequence a constructive strong normalization proof for $T^{\star}$ is obtained. Another consequence is that every $T^{\star}$ -representable number-theoretic function is elementary (...) recursive. Furthermore, it is shown that, conversely, every elementary recursive function is representable in $T^{\star}$ .The expressive weakness of $T^{\star}$ compared to the full system T can be explained as follows: In contrast to $T$ , computation steps in $T^{\star}$ never increase the nesting-depth of ${\mathcal I}_\rho$ and ${\mathcal R}_\rho$ at recursion positions. (shrink)

is a fragment of first-order aritlimetic so weak that it cannot prove the totality of an iterated exponential fimction. Surprisingly, however, the theory is remarkably robust. I will discuss formal results that show that many theorems of number theory and combinatorics are derivable in elementary arithmetic, and try to place these results in a broader philosophical context.