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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.
The µ Quantification Operator in Explicit Mathematics With Universes and Iterated Fixed Point Theories With Ordinals
, 1998
"... This paper is about two topics: 1. systems of explicit mathematics with universes and a nonconstructive quantification operator ¯; 2. iterated fixed point theories with ordinals. We give a prooftheoretic treatment of both families of theories; in particular, ordinal theories are used to get upper ..."
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Cited by 5 (3 self)
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This paper is about two topics: 1. systems of explicit mathematics with universes and a nonconstructive quantification operator ¯; 2. iterated fixed point theories with ordinals. We give a prooftheoretic treatment of both families of theories; in particular, ordinal theories are used to get upper bounds for explicit theories with finitely many universes. 1 Introduction The two major frameworks for explicit mathematics that were introduced in Feferman [4, 5] are the theories T 0 and T 1 . T 1 results from T 0 by strengthening the applicative axioms by the socalled nonconstructive ¯ operator. Although highly nonconstructive, ¯ is predicatively acceptable and makes quantification over the natural numbers explicit. While the proof theory of T 0 is wellknown since the early eighties (cf. Feferman [4, 5], Feferman and Sieg [10], Jager [14], Jager and Pohlers [17]), the corresponding investigations of subystems of T 1 have been completed only recently by Feferman and Jager [9, 8] and G...
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 ...
Σ 1 1 choice in a theory of sets and classes
"... Dedicated to Wolfram Pohlers on his retirement Several decades ago Friedman showed that the subsystem Σ1 1AC of second order arithmetic is prooftheoretically equivalent – and thus equiconsistent – to (Π1 0CA)<ε0. In this article we prove the analogous result for Σ1 1 choice in the context of the ..."
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Cited by 4 (2 self)
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Dedicated to Wolfram Pohlers on his retirement Several decades ago Friedman showed that the subsystem Σ1 1AC of second order arithmetic is prooftheoretically equivalent – and thus equiconsistent – to (Π1 0CA)<ε0. In this article we prove the analogous result for Σ1 1 choice in the context of the von NeumannBernaysGödel theory NBG of sets and classes.
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
Systems of explicit mathematics with nonconstructive µoperator and join
 ANNALS OF PURE AND APPLIED LOGIC
, 1996
"... The aim of this article is to give the prooftheoretic analysis of various subsystems of Feferman's theory T1 for explicit mathematics which contain the nonconstructive µoperator and join. We make use of standard prooftheoretic techniques such as cutelimination of appropriate semiformal systems ..."
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Cited by 2 (2 self)
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The aim of this article is to give the prooftheoretic analysis of various subsystems of Feferman's theory T1 for explicit mathematics which contain the nonconstructive µoperator and join. We make use of standard prooftheoretic techniques such as cutelimination of appropriate semiformal systems and asymmetrical interpretations in standard structures for explicit mathematics.
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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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. 1 Introduction The formal system of classical analysis is second order arithmetic with the full comprehension principle. It was baptized classical analysis, since classical mathematical analysis can be formalized in it. Often, subsystems of classical analysis su#ce as formal framework for particular parts of mathematical analysis. During the last decades a lot of such subsystems have been isolated and prooftheoretically investigated. The subsystems of analysis introduced in this paper belong to metapredicative prooftheory. Metapredicative systems have prooftheoretic ordinals beyond # 0 but can still be tr...
Explicit Mathematics With The Monotone Fixed Point Principle. II: Models
 Journal of Symbolic Logic
, 1999
"... This paper continues investigations of the monotone fixed point principle in the context of Feferman's explicit mathematics begun in [14]. Explicit mathematics is a versatile formal framework for representing Bishopstyle constructive mathematics and generalized recursion theory. The object of inves ..."
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This paper continues investigations of the monotone fixed point principle in the context of Feferman's explicit mathematics begun in [14]. Explicit mathematics is a versatile formal framework for representing Bishopstyle constructive mathematics and generalized recursion theory. The object of investigation here is the theory of explicit mathematics augmented by the monotone fixed point principle, which asserts that any monotone operation on classifications (Feferman's notion of set) possesses a least fixed point. To be more precise, the new axiom not merely postulates the existence of a least solution, but, by adjoining a new constant to the language, it is ensured that a fixed point is uniformly presentable as a function of the monotone operation. Let T 0 +UMID denote this extension of explicit mathematics. [14] gave lower bounds for the strength of two subtheories of T 0 +UMID in relating them to fragments of second order arithmetic based on \Pi 1 2 comprehension. [14] showed that ...
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.