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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.
Polynomial Time Operations in Explicit Mathematics
 Journal of Symbolic Logic
, 1997
"... 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 selfapplication and whose provably total functions on W = f0; 1g are exactly the polynomial time computable fu ..."
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Cited by 7 (5 self)
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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 selfapplication and whose provably total functions on W = f0; 1g are exactly the polynomial time computable functions.
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.
Natural Numbers and Forms of Weak Induction in Applicative Theories
, 1995
"... 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 Nstrictness, we give a natural extension of the theory BON which implies the equivalence of operati ..."
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Cited by 2 (0 self)
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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 Nstrictness, we give a natural extension of the theory BON which implies the equivalence of operation and Ninduction. In addition, we show that in the presence of the nonconstructive ¯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 socalled nonconstructive ¯operator, a predicatively acceptable ...
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
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