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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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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...
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...
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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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...
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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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 ...
The prooftheoretic analysis of Σ 1 1 transfinite dependent choice
 Annals of Pure and Applied Logic 121 (2003
"... choice. This article provides an ordinal analysis of Σ 1 1 transfinite dependent 1 ..."
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choice. This article provides an ordinal analysis of Σ 1 1 transfinite dependent 1
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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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
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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. 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...
Intuitionistic Fixed Point Theories for Strictly Positive Operators
, 2001
"... In this paper it is shown that the intuitionistic fixed point theory # ID i # (strict) for # times iterated fixed points of strictly positive operator forms is conservative for negative arithmetic and # 0 2 sentences over the theory ACA i # for # times iterated arithmetic comprehension without ..."
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In this paper it is shown that the intuitionistic fixed point theory # ID i # (strict) for # times iterated fixed points of strictly positive operator forms is conservative for negative arithmetic and # 0 2 sentences over the theory ACA i # for # times iterated arithmetic comprehension without set parameters. This generalizes results previously due to Buchholz [5] and Arai [2].