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Totality in Applicative Theories
 ANNALS OF PURE AND APPLIED LOGIC
, 1995
"... In this paper we study applicative theories of operations and numbers with (and without) the nonconstructive minimum operator in the context of a total application operation. We determine the prooftheoretic strength of such theories by relating them to wellknown systems like Peano Arithmetic ..."
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Cited by 21 (12 self)
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In this paper we study applicative theories of operations and numbers with (and without) the nonconstructive minimum operator in the context of a total application operation. We determine the prooftheoretic strength of such theories by relating them to wellknown systems like Peano Arithmetic PA and the system (\Pi 0 1 CA) !"0 of second order arithmetic. Essential use will be made of socalled fixedpoint theories with ordinals, certain infinitary term models and Church Rosser properties.
Some Theories With Positive Induction of Ordinal Strength ...
 JOURNAL OF SYMBOLIC LOGIC
, 1996
"... 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 ord ..."
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Cited by 7 (3 self)
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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 prooftheoretic strength '!0.
The µ Quantification Operator in Explicit Mathematics With Universes and Iterated Fixed Point Theories With Ordinals
, 1998
"... This paper is about two topics: 1. systems of explicit mathematics with universes and a nonconstructive quantification operator ¯; 2. iterated fixed point theories with ordinals. We give a prooftheoretic treatment of both families of theories; in particular, ordinal theories are used to get upper ..."
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Cited by 5 (3 self)
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This paper is about two topics: 1. systems of explicit mathematics with universes and a nonconstructive quantification operator ¯; 2. iterated fixed point theories with ordinals. We give a prooftheoretic 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 socalled nonconstructive ¯ operator. Although highly nonconstructive, ¯ is predicatively acceptable and makes quantification over the natural numbers explicit. While the proof theory of T 0 is wellknown 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 NonFinitist Arithmetic
, 2000
"... The unfolding of schematic formal systems is a novel concept which was initiated in Feferman [6]. This paper is mainly concerned with the prooftheoretic analysis of various unfolding systems for nonnitist arithmetic NFA. In particular, we examine two restricted unfoldings U 0 (NFA) and U 1 (NFA ..."
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Cited by 5 (3 self)
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The unfolding of schematic formal systems is a novel concept which was initiated in Feferman [6]. This paper is mainly concerned with the prooftheoretic analysis of various unfolding systems for nonnitist 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 prooftheoretically equivalent to predicative analysis.
Systems of explicit mathematics with nonconstructive µoperator and join
 ANNALS OF PURE AND APPLIED LOGIC
, 1996
"... The aim of this article is to give the prooftheoretic analysis of various subsystems of Feferman's theory T1 for explicit mathematics which contain the nonconstructive µoperator and join. We make use of standard prooftheoretic techniques such as cutelimination of appropriate semiformal systems ..."
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Cited by 2 (2 self)
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The aim of this article is to give the prooftheoretic analysis of various subsystems of Feferman's theory T1 for explicit mathematics which contain the nonconstructive µoperator and join. We make use of standard prooftheoretic techniques such as cutelimination of appropriate semiformal systems and asymmetrical interpretations in standard structures for explicit mathematics.
The NonConstructive µ Operator, Fixed Point Theories With Ordinals, and the Bar Rule
, 2000
"... This paper deals with the proof theory of first order applicative theories with nonconstructive 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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Cited by 2 (2 self)
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This paper deals with the proof theory of first order applicative theories with nonconstructive 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.
On the Proof Theory of Applicative Theories
 PHD THESIS, INSTITUT FÜR INFORMATIK UND ANGEWANDTE MATHEMATIK, UNIVERSITÄT
, 1996
"... ..."
Fixed Point Theories and Dependent Choice
, 2000
"... In this paper we establish the prooftheoretic equivalence of (i) ATR and c ID ! , (ii) ATR 0 + (\Sigma 1 1 DC) and c ID !! ! , and (iii) ATR + (\Sigma 1 1 DC) and c ID !"0 . ..."
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Cited by 1 (0 self)
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In this paper we establish the prooftheoretic equivalence of (i) ATR and c ID ! , (ii) ATR 0 + (\Sigma 1 1 DC) and c ID !! ! , and (iii) ATR + (\Sigma 1 1 DC) and c ID !"0 .