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 (\Pi 0 1 -CA) !"0 of second order arithmetic. Essential use will be made of so-called fixed-point theories with ordinals, certain infinitary term models and Church Rosser properties.

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.

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.

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 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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