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On universes in type theory
 191 – 204
, 1998
"... The notion of a universe of types was introduced into constructive type theory by MartinLöf (1975). According to the propositionsastypes principle inherent in ..."
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Cited by 35 (8 self)
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The notion of a universe of types was introduced into constructive type theory by MartinLöf (1975). According to the propositionsastypes principle inherent in
Inaccessibility in Constructive Set Theory and Type Theory
, 1998
"... This paper is the first in a series whose objective is to study notions of large sets in the context of formal theories of constructivity. The two theories considered are Aczel's constructive set theory (CZF) and MartinLof's intuitionistic theory of types. This paper treats Mahlo&apos ..."
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Cited by 20 (5 self)
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This paper is the first in a series whose objective is to study notions of large sets in the context of formal theories of constructivity. The two theories considered are Aczel's constructive set theory (CZF) and MartinLof's intuitionistic theory of types. This paper treats Mahlo's numbers which give rise classically to the enumerations of inaccessibles of all transfinite orders. We extend the axioms of CZF and show that the resulting theory, when augmented by the tertium non datur, is equivalent to ZF plus the assertion that there are inaccessibles of all transfinite orders. Finally the theorems of that extension of CZF are interpreted in an extension of MartinLof's intuitionistic theory of types by a universe. 1 Prefatory and historical remarks The paper is organized as follows: After recalling Mahlo's numbers and relating the history of universes in MartinLof type theory in section 1, we study notions of inaccessibility in the context of Aczel's constructive set theo...
On the relationship between ATR0 and ÎD<ω
 J. Symb. Logic
, 1996
"... We show that the theory ATR0 is equivalent to a secondorder generalization of the theory cID<ω. As a result, ATR0 is conservative over cID<ω for arithmetic sentences, though proofs in ATR0 can be much shorter than their cID<ω counterparts. 1 ..."
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We show that the theory ATR0 is equivalent to a secondorder generalization of the theory cID<ω. As a result, ATR0 is conservative over cID<ω for arithmetic sentences, though proofs in ATR0 can be much shorter than their cID<ω counterparts. 1
A ModelTheoretic Approach to Ordinal Analysis
 Bulletin of Symbolic Logic
, 1997
"... . We describe a modeltheoretic approach to ordinal analysis via the finite combinatorial notion of an #large set of natural numbers. In contrast to syntactic approaches that use cut elimination, this approach involves constructing finite sets of numbers with combinatorial properties that, in no ..."
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Cited by 12 (3 self)
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. We describe a modeltheoretic approach to ordinal analysis via the finite combinatorial notion of an #large set of natural numbers. In contrast to syntactic approaches that use cut elimination, this approach involves constructing finite sets of numbers with combinatorial properties that, in nonstandard instances, give rise to models of the theory being analyzed. This method is applied to obtain ordinal analyses of a number of interesting subsystems of first and secondorder arithmetic. x1. Introduction. Two of proof theory's defining goals are the justification of classical theories on constructive grounds, and the extraction of constructive information from classical proofs. Since Gentzen, ordinal analysis has been a major component in these pursuits, and the assignment of recursive ordinals to theories has proven to be an illuminating way of measuring their constructive strength. The traditional approach to ordinal analysis, which uses cutelimination procedures to transfor...
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...
Weyl’s predicative classical mathematics as a logicenriched type theory
, 2006
"... Abstract. In Das Kontinuum, Weyl showed how a large body of classical mathematics could be developed on a purely predicative foundation. We present a logicenriched type theory that corresponds to Weyl’s foundational system. A large part of the mathematics in Weyl’s book — including Weyl’s definitio ..."
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Abstract. In Das Kontinuum, Weyl showed how a large body of classical mathematics could be developed on a purely predicative foundation. We present a logicenriched type theory that corresponds to Weyl’s foundational system. A large part of the mathematics in Weyl’s book — including Weyl’s definition of the cardinality of a set and several results from real analysis — has been formalised, using the proof assistant Plastic that implements a logical framework. This case study shows how type theory can be used to represent a nonconstructive foundation for mathematics. Key words: logicenriched type theory, predicativism, formalisation 1
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 8 (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 AntiFoundation Axiom In Constructive Set Theories
 Stanford University Press
, 2003
"... . The paper investigates the strength of the antifoundation axiom on the basis of various systems of constructive set theories. 1. Introduction Intrinsically circular phenomena have come to the attention of researchers in differing fields such as mathematical logic, computer science, artificial inte ..."
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. The paper investigates the strength of the antifoundation axiom on the basis of various systems of constructive set theories. 1. Introduction Intrinsically circular phenomena have come to the attention of researchers in differing fields such as mathematical logic, computer science, artificial intelligence, linguistics, cognitive science, and philosophy. Logicians first explored set theories whose universe contains what are called nonwellfounded sets, or hypersets (cf. [17], [5]). But the area was considered rather exotic until these theories were put to use in developing rigorous accounts of circular notions in computer science (cf. [7]). Instead of the Foundation Axiom these set theories adopt the socalled AntiFoundation Axiom, AFA, which gives rise to a rich universe of sets. AFA provides an elegant tool for modeling all sorts of circular phenomena. The application areas range from knowledge representation and theoretical economics to the semantics of natural language and pr...
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...
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 µ; 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 µ; 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.