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)

We establish by elementary proof-theoretic means the conservativeness of two subsystems of analysis over primitive recursive arithmetic. The one subsystem was introduced by Friedman [6], the other is a strengthened version of a theory of Minc [14]; each has been shown to be of considerable interest for both mathematical practice and metamathematical investigations. The foundational significance of such conservation results is clear: they provide a direct finitist justification of the part of mathematical practice formalizable in these subsystems. The results are (...) generalized to relate a hierarchy of subsystems, all contained in the theory of arithmetic properties, to a corresponding hierarchy of fragments of arithmetic. The proof theoretic tools employed there are used to re-establish in a uniform, elementary way relationships between various fragments of arithmetic due to Parsons, Paris and Kirby, and Friedman. (shrink)

LetA H be the Herbrand normal form ofA andA H,D a Herbrand realization ofA H. We showThere is an example of an (open) theory ℐ+ with function parameters such that for someA not containing function parameters Similar for first order theories ℐ+ if the index functions used in definingA H are permitted to occur in instances of non-logical axiom schemata of ℐ, i.e. for suitable ℐ,A In fact, in (1) we can take for ℐ+ the fragment (Σ 1 0 -IA)+ (...) of second order arithmetic with induction restricted toΣ 1 0 -formulas, and in (2) we can take for ℐ the fragment (Σ 1 0,b -IA) of first order arithmetic with induction restricted to formulas VxA(x) whereA contains only bounded quantifiers.On the other hand, $$PA^2 \vdash A^H \Rightarrow PA \vdash A,$$ wherePA 2 is the extension of first order arithmeticPA obtained by adding quantifiers for functions andA∈ℒ(PA). This generalizes to extensional arithmetic in the language of all finite types but not to sentencesA with positively occurring existential quantifiers for functions. (shrink)

A pointwise version of the Howard-Bezem notion of hereditary majorization is introduced which has various advantages, and its relation to the usual notion of majorization is discussed. This pointwise majorization of primitive recursive functionals (in the sense of Gödel'sT as well as Kleene/Feferman's ) is applied to systems of intuitionistic and classical arithmetic (H andH c) in all finite types with full induction as well as to the corresponding systems with restricted inductionĤ↾ andĤ↾c.H and Ĥ↾ are closed under a generalized (...) fan-rule. For a restricted class of formulae this also holds forH c andĤ↾c.We give a new and very perspicuous proof that for each one can construct a functional such that $\tilde \Phi \alpha $ is a modulus of uniform continuity for Φ on {β1∣∧n(βn≦αn)}. Such a modulus can also be obtained by majorizing any modulus of pointwise continuity for Φ.The type structure ℳ of all pointwise majorizable set-theoretical functionals of finite type is used to give a short proof that quantifier-free “choice” with uniqueness (AC!)1,0-qf. is not provable within classical arithmetic in all finite types plus comprehension [given by the schema (C)ϱ:∨y 0ϱ∧x ϱ(yx=0⇔A(x)) for arbitraryA], dependent ω-choice and bounded choice. Furthermore ℳ separates several μ-operators. (shrink)

ZusammenfassungP. Bernays hat darauf hingewiesen, dass man, um die Widerspruchs freiheit der klassischen Zahlentheorie zu beweisen, den Hilbertschen flniter Standpunkt dadurch erweitern muss, dass man neben den auf Symbole sich beziehenden kombinatorischen Begriffen gewisse abstrakte Begriffe zulässt, Die abstrakten Begriffe, die bisher für diesen Zweck verwendet wurden, sinc die der konstruktiven Ordinalzahltheorie und die der intuitionistischer. Logik. Es wird gezeigt, dass man statt deesen den Begriff einer berechenbaren Funktion endlichen einfachen Typs über den natürlichen Zahler benutzen kann, wobei keine anderen (...) Konstruktionsverfahren für solche Funktionen nötig sind, als einfache Rekursion nach einer Zahlvariablen und Einsetzung von Funktionen ineinander .P. Bernays has pointed out that, in order to prove the consistency of classical number theory, it is necessary to extend Hilbert's finitary stand‐point by admitting certain abstract concepts in addition to the combinatorial concepts referring to symbols. The abstract concepts that so far have been used for this purpose are those of the constructive theory of ordinals and those of intuitionistic logic. It is shown that the concept of a computable function of finite simple type over the integers can be used instead, where no other procedures of constructing such functions are necessary except simple recursion by an integral variable and substitution of functions in each other. (shrink)

IΣ n andBΣ n are well known fragments of first-order arithmetic with induction and collection forΣ n formulas respectively;IΣ n 0 andBΣ n 0 are their second-order counterparts. RCA0 is the well known fragment of second-order arithmetic with recursive comprehension;WKL 0 isRCA 0 plus weak König's lemma. We first strengthen Harrington's conservation result by showing thatWKL 0 +BΣ n 0 is Π 1 1 -conservative overRCA 0 +BΣ n 0 . Then we develop some model theory inWKL 0 and illustrate (...) the use of formalized model theory by giving a relatively simple proof of the fact thatIΣ 1 provesBΣ n+1 to be Π n+2-conservative overIΣ n . Finally, we present a proof-theoretic proof of the stronger fact that theΠ n+2 conservation result is provable already inIΔ 0 + superexp. ThusIΣ n+1 proves 1-Con (BΣ n+1) andIΔ 0 +superexp proves Con (IΣ n )↔Con(BΣ n+1). (shrink)