In this paper we discuss extensions of Feferman's theory T_0 for explicit mathematics by the so-called limit and Mahlo axioms and present a novel approach to constructing natural recusion-theoretic models for (fairly strong) systems of explicit mathematics which is based on nonmonotone inductive definitions.

Applicative theories form the basis of Feferman's systems of explicit mathematics, which have been introduced in the early seventies. 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 deals with universes in explicit mathematics. After introducing some basic definitions, the limit axiom and possible ordering principles for universes are discussed. Later, we turn to least universes, strictness and name induction. Special emphasis is put on theories for explicit mathematics with universes which are proof-theoretically equivalent to Feferman's T 0 . 1 Introduction In some form or another, universes play an important role in many systems of set theory and higher order arithmetic, in various formalizations of constructive mathematics and in logics for computation. One aspect of universes is that they expand the set or type formation principles in a natural and perspicuous way and provide greater expressive power and proof-theoretic strength. The general idea behind universes is quite simple: suppose that we are given a formal system Th comprising certain set (or type) existence principles which are justified on specific philosophical grounds. Then it may be a...

This paper studies systems of explicit mathematics as introduced by Feferman [9, 11]. In particular, we propose weak explicit type systems with a restricted form of elementary comprehension whose provably terminating operations coincide with the functions on binary words that are computable in polynomial time. The systems considered are natural extensions of the first-order applicative theories introduced in

In Spescha and Strahm [15], a system PET of explicit mathematics in the style of Feferman [7, 8] is introduced, and in Spescha and Strahm [16] the addition of the join principle to PET is studied. Changing to intuitionistic logic, it could be shown that the provably terminating operations of PETJ i are the polytime functions on binary words. However, although strongly conjectured, it remained open whether the same holds true for the corresponding theory PETJ with classical logic. This note supplements a proof of this conjecture. Keywords: Explicit mathematics, polytime functions, non-standard models

This is a survey paper on various weak systems of Feferman’s explicit mathematics and their proof theory. The strength of the systems considered in measured in terms of their provably terminating operations typically belonging to some natural classes of computational time or space complexity. Keywords: Proof theory, Feferman’s explicit mathematics, applicative theories, higher types, types and names, partial truth, feasible operations 1

In this paper we discuss extensions of Feferman's theory T 0 for explicit mathematics by the so-called limit and Mahlo axioms and present a novel approach to constructing natural recusion-theoretic models for (fairly strong) systems of explicit mathematics which is based on nonmonotone inductive definitions. 1 Introduction The purpose of this paper is twofold: our first aim is to discuss the extensions of Feferman's theory T 0 for explicit mathematics by the so-called limit and Mahlo axioms; secondly, we want to present a novel approach to constructing natural recusion-theoretic models for (fairly strong) systems of explicit mathematics which is based on nonmonotone inductive definitions. Subsystems of second order arithmetic and set theory dealing with or describing a recursively inaccessible or Mahlo universe play an important role in proof theory for quite some time. In this context we have to mention the theories (# 1 2 -CA) + (BI) and KPi (cf. e.g. Jager [8]) whose least standar...

The aim of this article is to give the proof-theoretic analysis of various subsystems of Feferman's theory T1 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 semi-formal systems and asymmetrical interpretations in standard structures for explicit mathematics.

This paper continues investigations of the monotone fixed point principle in the context of Feferman's explicit mathematics begun in [14]. Explicit mathematics is a versatile formal framework for representing Bishop-style constructive mathematics and generalized recursion theory. The object of investigation here is the theory of explicit mathematics augmented by the monotone fixed point principle, which asserts that any monotone operation on classifications (Feferman's notion of set) possesses a least fixed point. To be more precise, the new axiom not merely postulates the existence of a least solution, but, by adjoining a new constant to the language, it is ensured that a fixed point is uniformly presentable as a function of the monotone operation. Let T 0 +UMID denote this extension of explicit mathematics. [14] gave lower bounds for the strength of two subtheories of T 0 +UMID in relating them to fragments of second order arithmetic based on \Pi 1 2 comprehension. [14] showed that ...

Church introduced -calculus in the beginning of the thirties as a foundation of mathematics and map theory from around 1992 fulfilled that primary aim. The present paper presents a new version of map theory whose axioms are simpler and better motivated than those of the original version from 1992. The paper focuses on the semantics of map theory and explains this semantics on basis of -Scott domains. The new version sheds some light on the difference between Russells and Burali-Fortis paradoxes, and also sheds some light on why it is consistent to allow non-well-founded sets in a ZF-style system. DIKU, University of Copenhagen, Universitetsparken 1, DK-2100 Copenhagen, Denmark, E-mail grue@diku.dk Contents 1 Introduction 3 1.1 Differences with Churchs approach . . . . . . . . . . . . . . . . . 4 1.2 Inclusion of non-functions . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Set abstraction versus -abstraction . . . . . . . . . . . . . . . . 5 1.4 Selection of well-behaved maps ....