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39
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.
Upper bounds for metapredicative Mahlo in explicit mathematics and admissible set theory
 Journal of Symbolic Logic
"... In this article we introduce systems for metapredicative Mahlo in explicit mathematics and admissible set theory. The exact upper prooftheoretic bounds of these systems are established. 1 Introduction In classical set theory an ordinal # is called a Mahlo ordinal if it is a regular cardinal and if, ..."
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Cited by 20 (14 self)
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In this article we introduce systems for metapredicative Mahlo in explicit mathematics and admissible set theory. The exact upper prooftheoretic bounds of these systems are established. 1 Introduction In classical set theory an ordinal # is called a Mahlo ordinal if it is a regular cardinal and if, for every normal function f from # to #, there exists a regular cardinal less than # so that {f(#) : # < } # . The statement that there exists a Mahlo ordinal is a powerful set existence axiom going beyond theories like ZFC. It also outgrows the existence of inaccessible cardinals, hyper inaccessibles, hyperhyperinaccessible and the like. There is also an obvious recursive analogue of Mahlo ordinal. Typically, an ordinal # is baptized recursively Mahlo, if it is admissible and reflects every # 2 sentence on a smaller admissible ordinal. The corresponding formal theory KPM has been prooftheoretically analyzed by Rathjen [14, 15]. KPM is a highly impredicative theory, and its proofthe...
Extending the System T_0 of explicit mathematics: the limit and Mahlo axioms
"... In this paper we discuss extensions of Feferman's theory T_0 for explicit mathematics by the socalled limit and Mahlo axioms and present a novel approach to constructing natural recusiontheoretic models for (fairly strong) systems of explicit mathematics which is based on nonmonotone inductive def ..."
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Cited by 13 (8 self)
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In this paper we discuss extensions of Feferman's theory T_0 for explicit mathematics by the socalled limit and Mahlo axioms and present a novel approach to constructing natural recusiontheoretic models for (fairly strong) systems of explicit mathematics which is based on nonmonotone inductive definitions.
Theories With SelfApplication and Computational Complexity
 Information and Computation
, 2002
"... 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: selfapplication is meaningful, but not ne ..."
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Cited by 12 (9 self)
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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: selfapplication is meaningful, but not necessarily total. It has turned out that theories with selfapplication provide a natural setting for studying notions of abstract computability, especially from a prooftheoretic perspective.
Foundational and mathematical uses of higher types
 REFLECTIONS ON THE FOUNDATIONS OF MATHEMATICS: ESSAY IN HONOR OF SOLOMON FEFERMAN
, 1999
"... In this paper we develop mathematically strong systems of analysis in higher types which, nevertheless, are prooftheoretically weak, i.e. conservative over elementary resp. primitive recursive arithmetic. These systems are based on noncollapsing hierarchies ( n WKL+ ; n WKL+ ) of principles ..."
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Cited by 11 (4 self)
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In this paper we develop mathematically strong systems of analysis in higher types which, nevertheless, are prooftheoretically weak, i.e. conservative over elementary resp. primitive recursive arithmetic. These systems are based on noncollapsing hierarchies ( n WKL+ ; n WKL+ ) of principles which generalize (and for n = 0 coincide with) the socalled `weak' König's lemma WKL (which has been studied extensively in the context of second order arithmetic) to logically more complex tree predicates. Whereas the second order context used in the program of reverse mathematics requires an encoding of higher analytical concepts like continuous functions F : X ! Y between Polish spaces X;Y , the more exible language of our systems allows to treat such objects directly. This is of relevance as the encoding of F used in reverse mathematics tacitly yields a constructively enriched notion of continuous functions which e.g. for F : IN ! IN can be seen (in our higher order context)
Universes in Explicit Mathematics
 Annals of Pure and Applied Logic
, 1999
"... 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 mathemat ..."
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Cited by 8 (5 self)
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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 prooftheoretically 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 prooftheoretic 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...
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.
Wellordering proofs for metapredicative Mahlo
 Journal of Symbolic Logic
"... In this article we provide wellordering proofs for metapredicative systems of explicit mathematics and admissible set theory featuring suitable axioms about the Mahloness of the underlying universe of discourse. In particular, it is shown that in the corresponding theories EMA of explicit mathemati ..."
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Cited by 6 (1 self)
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In this article we provide wellordering proofs for metapredicative systems of explicit mathematics and admissible set theory featuring suitable axioms about the Mahloness of the underlying universe of discourse. In particular, it is shown that in the corresponding theories EMA of explicit mathematics and KPm 0 of admissible set theory, transfinite induction along initial segments of the ordinal ##00, for # being a ternary Veblen function, is derivable. This reveals that the upper bounds given for these two systems in the paper Jager and Strahm [11] are indeed sharp. 1 Introduction This paper is a companion to the article Jager and Strahm [11], where systems of explicit mathematics and admissible set theory for metapredicative Mahlo are introduced. Whereas the main concern of [11] was to establish prooftheoretic upper bounds for these systems, in this article we provide the corresponding wellordering proofs, thus showing that the upper bounds derived in [11] are sharp. The central...
Elementary explicit types and polynomial time operations
, 2008
"... 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 polyn ..."
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Cited by 6 (5 self)
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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 firstorder applicative theories introduced in