Thomas Strahm

Upper Bounds for Metapredicative Mahlo in Explicit Mathematics and Admissible Set Theory

In this article we introduce systems for metapredicative Mahlo in explicit mathematics and admissible set theory. The exact upper prooftheoretic bounds of these systems are established.

Totality in applicative theories

Abstract In this paper we study applicative theories of operations and numbers with (and without) the non-constructive minimum operator in the context of a total application operation. We determine the proof-theoretic strength of such theories by relating them to well-known systems like Peano Arithmetic PA and the system (Π∞0-CA)

The proof-theoretic analysis of transfinitely iterated fixed point theories

This article provides the proof-theoretic analysis of the transfinitely iterated fixed point theories (alpha) and (<alpha); the exact proof-theoretic ordinals of these systems are presented.

The unfolding of non-finitist arithmetic

The unfolding of schematic formal systems is a novel concept which was initiated in Feferman (in: Hajek (Ed.), Godel ’96, Lecture Notes in Logic, Springer, Berlin, 1996, pp. 3–22). This paper is mainly concerned with the proof-theoretic analysis of various unfolding systems for non-finitist arithmetic NFA. In particular, we examine two restricted unfoldings U0(NFA) and U1(NFA), as well as a full unfolding, U(NFA). The principal results then state: (i) U0(NFA) is equivalent to PA; (ii) U1(NFA) is equivalent to RA<ω; (iii) U(NFA) is equivalent to RA<Γ0. Thus U(NFA) is proof-theoretically equivalent to predicative analysis.

On Applicative Theories

Systems of explicit mathematics were introduced in Feferman [7, 9] in order to give a logical account to Bishop-style constructive mathematics, and they soon turned out to be very important for the proof-theoretic analysis of subsystems of second order arithmetic and set theory. Moreover, systems of explicit mathematics provide a logical framework for functional programming languages.

Theories with self-application and computational complexity

Applicative theories form the basis of Fefermans systems of explicit mathematics, which have been introduced in the 1970s. In an applicative universe, all individuals may be thought of as operations, which can freely be applied to each other: self-application is meaningful, but not necessarily total. It has turned out that theories with self-application provide a natural setting for studying notions of abstract computability, especially from a proof-theoretic perspective. This paper is concerned with the study of (unramified) bounded applicative theories which have a strong relationship to classes of computational complexity. We propose new applicative systems whose provably total functions coincide with the functions computable in polynomial time, polynomial space, polynomial time and linear space, as well as linear space. Our theories can be regarded as applicative analogues of traditional systems of bounded arithmetic. We are also interested in higher-type features of our systems; in particular, it is shown that Cook and Urquharts system PVω is directly contained in a natural applicative theory of polynomial strength.

Some theories with positive induction of ordinal strength φω 0

This paper deals with: (i) the theory ID*# which results ID1 from by restricting induction on the natural numbers to formulas which are positive in the fixed point constants, (ii) the theory BON(μ) plus various forms of positive induction, and (iii) a subtheory of Peano arithmetic with ordinals in which induction on the natural numbers is restricted to formulas which are Σ in the ordinals. We show that these systems have proof-theoretic strength φω0.

Systems of explicit mathematics with non-constructive μ-operator and join

Abstract The aim of this article is to give the proof-theoretic analysis of various subsystems of Fefermans theory T 1 for explicit mathematics which contain the non-constructive μ-operator and join. We make use of standard proof-theoretic techniques such as cut-elimination of appropriate semiformal systems and asymmetrical interpretations in standard structures for explicit mathematics.

First Steps into Metapredicativity in Explicit Mathematics

Metapredicativity is a new general term in proof theory which describes the analysis and study of formal systems whose proof-theoretic strength is beyond the Feferman-Schutte ordinal Γ0 but which are nevertheless amenable to purely predicative methods. Typical examples of formal systems which are apt for scaling the initial part of metapredicativity are the transfinitely iterated fixed point theories IDα whose detailed proof-theoretic analysis is given by Jager, Kahle, Setzer and Strahm in [18]. In this paper we assume familiarity with [18]. For natural extensions of Friedman’s ATR that can be measured against transfinitely iterated fixed point theories the reader is referred to Jager and Strahm [20]. In the mid seventies, Feferman [3, 4] introduced systems of explicit mathematics in order to provide an alternative foundation of constructive mathematics. More precisely, it was the origin of Feferman’s program to give a logical account of Bishop-style constructive mathematics. Right from the beginning, systems of explicit mathematics turned out to be of general interest for proof theory, mainly in connection with the proof-theoretic analysis of subsystems of first and second order arithmetic and set theory, cf. e.g. Jager [15] and Jager and Pohlers [19]. More recently, systems of explicit mathematics have been used to develop a general logical framework for functional programming and type theory, where it is possible to derive correctness and termination properties of functional programs. Important references in this connection are Feferman [6, 7, 9] and Jager [17]. Universes are a frequently studied concept in constructive mathematics at least since the work of Martin-Lof, cf. e.g. Martin-Lof [23] or Palmgren [27] for

Partial Applicative Theories and Explicit Substitutions

Systems based on theories with partial self application are relevant to the formalization of constructive mathematics and as a logical basis for functional programming languages In the literature they are either presented in the form of partial combinatory logic or the partial calculus and sometimes these two approaches are erroneously considered to be equivalent In this paper we address some defects of the partial calculus as a constructive framework for partial functions In particular the partial calculus is not embeddable into partial combinatory logic and it lacks the standard recursion theoretic model The main reason is a concept of substitution which is not consistent with a strongly intensional point of view We design a weakening of the partial calculus which can be embedded into partial combinatory logic As a consequence the natural numbers with partial recursive function application are a model of our system The novel point will be the use of explicit substitutions which have previously been studied in the literature in connection with the implementation of functional programming languages