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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...
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...
Weak theories of truth and explicit mathematics. Submitted for publication. 19
"... We study weak theories of truth over combinatory logic and their relationship to weak systems of explicit mathematics. In particular, we consider two truth theories TPR and TPT of primitive recursive and feasible strength. The latter theory is a novel abstract truththeoretic setting which is able t ..."
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Cited by 2 (2 self)
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We study weak theories of truth over combinatory logic and their relationship to weak systems of explicit mathematics. In particular, we consider two truth theories TPR and TPT of primitive recursive and feasible strength. The latter theory is a novel abstract truththeoretic setting which is able to interpret expressive feasible subsystems of explicit mathematics. 1
Metapredicative Subsystems of Analysis
 PH.D. THESIS, INSTITUT FÜR INFORMATIK UND ANGEWANDTE MATHEMATIK, UNIVERISTÄT BERN, 2000. & EXPLICIT MAHLO 21
, 2001
"... In this paper we present some metapredicative subsystems of analysis. We deal with reflection principles, #model existence axioms (limit axioms) and axioms asserting the existence of hierarchies. We show several equivalences of the introduced subsystems. In particular we prove the equivalence of # ..."
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Cited by 2 (0 self)
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In this paper we present some metapredicative subsystems of analysis. We deal with reflection principles, #model existence axioms (limit axioms) and axioms asserting the existence of hierarchies. We show several equivalences of the introduced subsystems. In particular we prove the equivalence of # 1 1 transfinite dependent choice and # 1 2 reflection on #models of # 1 1 DC.
Formalizing NonTermination of Recursive Programs
 J. of Logic and Algebraic Programming
, 2001
"... In applicative theories the recursion theorem provides a term rec which solves recursive equations. However, it is not provable that a solution obtained by rec is minimal. In the present paper we introduce an applicative theory in which it is possible to dene a least xed point operator. Still, o ..."
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Cited by 1 (0 self)
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In applicative theories the recursion theorem provides a term rec which solves recursive equations. However, it is not provable that a solution obtained by rec is minimal. In the present paper we introduce an applicative theory in which it is possible to dene a least xed point operator. Still, our theory has a standard recursion theoretic interpretation. 1
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"... A symbolic approach to the state graph based analysis of highlevel Markov reward models Ein symbolischer Ansatz für die Zustandsgraphbasierte Analyse von ..."
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A symbolic approach to the state graph based analysis of highlevel Markov reward models Ein symbolischer Ansatz für die Zustandsgraphbasierte Analyse von
The ProofTheoretic Analysis of ... Transfinite Dependent Choice
, 2000
"... This article provides an ordinal analysis of # 1 1 transfinite dependent choice. 1 Introduction # 1 1 TDC 0 (# 1 1 Transfinite Dependent Choice) is a natural strengthening of # 1 1 DC 0 . Both are subsystems of analysis and assure the existence of implicitly # 1 1 definable sequences (of set ..."
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This article provides an ordinal analysis of # 1 1 transfinite dependent choice. 1 Introduction # 1 1 TDC 0 (# 1 1 Transfinite Dependent Choice) is a natural strengthening of # 1 1 DC 0 . Both are subsystems of analysis and assure the existence of implicitly # 1 1 definable sequences (of sets). In # 1 1 DC 0 , the length of these sequences is #, whereas in # 1 1 TDC 0 we can choose these sequences along an arbitrary wellordering. # 1 1 DC 0 has prooftheoretic strength ##0 (cf. [2] ), it is a predicative theory. On the other hand, the prooftheoretic strength of # 1 1 TDC 0 is ##00. The prooftheoretic analysis given in this article shows that # 1 1 TDC 0 is in fact metapredicative. If we add complete induction for arbitrary formulas, then the corresponding prooftheoretic ordinals are ## 0 0 and ## 0 00. The theory # 1 1 TDC 0 and its prooftheoretic analysis typically belong to the new area of socalled metapredicative prooftheory. Metapredicative systems have p...