This paper deals with: (i) the theory ID # 1 which results from c ID 1 by restricting induction on the natural numbers to formulas which are positive in the fixed point constants, (ii) the theory BON() 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 \Sigma in the ordinals. We show that these systems have proof-theoretic strength '!0.

In this paper we study (self-)applicative theories of operations and binary words in the context of polynomial time computability. We propose a first order theory PTO which allows full self-application and whose provably total functions on W = f0; 1g are exactly the polynomial time computable functions.

This paper is about two topics: 1. systems of explicit mathematics with universes and a non-constructive quantification operator ¯; 2. iterated fixed point theories with ordinals. We give a proof-theoretic treatment of both families of theories; in particular, ordinal theories are used to get upper bounds for explicit theories with finitely many universes. 1 Introduction The two major frameworks for explicit mathematics that were introduced in Feferman [4, 5] are the theories T 0 and T 1 . T 1 results from T 0 by strengthening the applicative axioms by the so-called non-constructive ¯ operator. Although highly non-constructive, ¯ is predicatively acceptable and makes quantification over the natural numbers explicit. While the proof theory of T 0 is well-known since the early eighties (cf. Feferman [4, 5], Feferman and Sieg [10], Jager [14], Jager and Pohlers [17]), the corresponding investigations of subystems of T 1 have been completed only recently by Feferman and Jager [9, 8] and G...

The unfolding of schematic formal systems is a novel concept which was initiated in Feferman [6]. This paper is mainly concerned with the proof-theoretic analysis of various unfolding systems for non-nitist arithmetic NFA. In particular, we examine two restricted unfoldings U 0 (NFA) and U 1 (NFA), as well as a full unfolding, U(NFA). The principal results then state: (i) U 0 (NFA) is equivalent to PA; (ii) U 1 (NFA) is equivalent to RA<! ; (iii) U(NFA) is equivalent to RA< 0 . Thus U(NFA) is proof-theoretically equivalent to predicative analysis.

In this paper we study the relationship between forms of weak induction in theories of operations and numbers. Therefore, we investigate the structure of the natural numbers. Introducing a concept of N-strictness, we give a natural extension of the theory BON which implies the equivalence of operation and N-induction. In addition, we show that in the presence of the non-constructive ¯-operator the above equivalence is provable without this extension. 1 Introduction Applicative theories go back to Feferman's systems of explicit mathematics introduced in [Fef75, Fef79]. They are based on the basic theory of operations and numbers BON which is introduced in [FJ93] as the classic version of Beeson's theory EON (cf. [Bee85]) without induction. Combined with various induction principles, applicative theories provide a natural framework for constructive mathematics and functional programming. If they are strengthened by the so-called non-constructive ¯-operator, a predicatively acceptable ...

This paper deals with the proof theory of first order applicative theories with non-constructive operator and a form of the bar rule, yielding systems of ordinal strength 0 and '20, respectively. Relevant use is made of fixed point theories with ordinals plus bar rule.

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