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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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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...
First Steps Into Metapredicativity in Explicit Mathematics
, 1999
"... The system EMU of explicit mathematics incorporates the uniform construction of universes. In this paper we give a prooftheoretic treatment of EMU and show that it corresponds to transfinite hierarchies of fixed points of positive arithmetic operators, where the length of these fixed point hierarc ..."
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Cited by 5 (2 self)
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The system EMU of explicit mathematics incorporates the uniform construction of universes. In this paper we give a prooftheoretic treatment of EMU and show that it corresponds to transfinite hierarchies of fixed points of positive arithmetic operators, where the length of these fixed point hierarchies is bounded by # 0 . 1 Introduction Metapredicativity is a new general term in proof theory which describes the analysis and study of formal systems whose prooftheoretic strength is beyond the FefermanSchutte ordinal # 0 but which are nevertheless amenable to purely predicative methods. Typical examples of formal systems which are apt for scaling the initial part of metapredicativity are the transfinitely iterated fixed point theories # ID # whose detailed prooftheoretic analysis is given by Jager, Kahle, Setzer and Strahm in [18]. In this paper we assume familiarity with [18]. For natural extensions of Friedman's ATR that can be measured against transfinitely iterated fixed point ...
Autonomous Fixed Point Progressions and Fixed Point Transfinite Recursion
 In Logic Colloquium ’98
"... . This paper is a contribution to the area of metapredicative proof theory. It continues recent investigations on the transfinitely iterated fixed point theories # ID# (cf. [10]) and addresses the question of autonomity in iterated fixed point theories. An external and an internal form of autonom ..."
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Cited by 3 (0 self)
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. This paper is a contribution to the area of metapredicative proof theory. It continues recent investigations on the transfinitely iterated fixed point theories # ID# (cf. [10]) and addresses the question of autonomity in iterated fixed point theories. An external and an internal form of autonomous generation of transfinite hierarchies of fixed points of positive arithmetic operators are introduced and prooftheoretically analyzed. This includes the discussion of the principle of socalled fixed point transfinite recursion. Connections to theories for iterated inaccessibility in the context of Kripke Platek set theory without foundation are revealed. 1 Introduction The foundational program to study the principles and ordinals which are implicit in a predicative conception of the universe of sets of natural numbers led to the progression of systems of ramified analysis up to the famous FefermanSchutte ordinal # 0 in the early sixties. Since then numerous theories have been found w...
Metapredicative And Explicit Mahlo: A ProofTheoretic Perspective
"... After briefly discussing the concepts of predicativity, metapredicativity and impredicativity, we turn to the notion of Mahloness as it is treated in various contexts. Afterwards the ..."
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Cited by 3 (2 self)
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After briefly discussing the concepts of predicativity, metapredicativity and impredicativity, we turn to the notion of Mahloness as it is treated in various contexts. Afterwards the
Reflections on reflections in explicit mathematics
 Ann. Pure Appl. Logic
, 2005
"... We give a broad discussion of reflection principles in explicit mathematics, thereby addressing various kinds of universe existence principles. The prooftheoretic strength of the relevant systems of explicit mathematics is couched in terms of suitable extensions of KripkePlatek set theory. 1 ..."
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Cited by 2 (1 self)
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We give a broad discussion of reflection principles in explicit mathematics, thereby addressing various kinds of universe existence principles. The prooftheoretic strength of the relevant systems of explicit mathematics is couched in terms of suitable extensions of KripkePlatek set theory. 1