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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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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.
Challenges to Predicative Foundations of Arithmetic
 in Between Logic and Intuition Essays in Honor of Charles Parsons
, 1996
"... This paper was written while the first author was a Fellow at the Center for Advanced Study in the Behavioral Sciences (Stanford, CA) whose facilities and support, under grants from the Andrew W. Mellon Foundation and the National Science Foundation, have been greatly appreciated. ..."
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Cited by 4 (0 self)
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This paper was written while the first author was a Fellow at the Center for Advanced Study in the Behavioral Sciences (Stanford, CA) whose facilities and support, under grants from the Andrew W. Mellon Foundation and the National Science Foundation, have been greatly appreciated.
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
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 2 (1 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. 1 Introduction The purpose of this paper is twofold: our first aim is to discuss the extensions of Feferman's theory T 0 for explicit mathematics by the socalled limit and Mahlo axioms; secondly, we want to present a novel approach to constructing natural recusiontheoretic models for (fairly strong) systems of explicit mathematics which is based on nonmonotone inductive definitions. Subsystems of second order arithmetic and set theory dealing with or describing a recursively inaccessible or Mahlo universe play an important role in proof theory for quite some time. In this context we have to mention the theories (# 1 2 CA) + (BI) and KPi (cf. e.g. Jager [8]) whose least standar...
Elementary constructive operational set theory. To appear in: Festschrift for Wolfram Pohlers, Ontos Verlag
"... Abstract. We introduce an operational set theory in the style of [5] and [17]. The theory we develop here is a theory of constructive sets and operations. One motivation behind constructive operational set theory is to merge a constructive notion of set ([1], [2]) with some aspects which are typical ..."
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Abstract. We introduce an operational set theory in the style of [5] and [17]. The theory we develop here is a theory of constructive sets and operations. One motivation behind constructive operational set theory is to merge a constructive notion of set ([1], [2]) with some aspects which are typical of explicit mathematics [14]. In particular, one has nonextensional operations (or rules) alongside extensional constructive sets. Operations are in general partial and a limited form of self–application is permitted. The system we introduce here is a fully explicit, finitely axiomatised system of constructive sets and operations, which is shown to be as strong as HA. 1.
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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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