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28
A finite axiomatization of inductiverecursive definitions
 Typed Lambda Calculi and Applications, volume 1581 of Lecture Notes in Computer Science
, 1999
"... Inductionrecursion is a schema which formalizes the principles for introducing new sets in MartinLöf’s type theory. It states that we may inductively define a set while simultaneously defining a function from this set into an arbitrary type by structural recursion. This extends the notion of an in ..."
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Cited by 42 (14 self)
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Inductionrecursion is a schema which formalizes the principles for introducing new sets in MartinLöf’s type theory. It states that we may inductively define a set while simultaneously defining a function from this set into an arbitrary type by structural recursion. This extends the notion of an inductively defined set substantially and allows us to introduce universes and higher order universes (but not a Mahlo universe). In this article we give a finite axiomatization of inductiverecursive definitions. We prove consistency by constructing a settheoretic model which makes use of one Mahlo cardinal. 1
An ObjectOriented Refinement Calculus with Modular Reasoning
, 1992
"... In this thesis, the refinement calculus is extended to support a variety of objectoriented programming styles. The late binding of procedure calls in objectoriented languages is modelled by defining an objectoriented system to be a function from procedure names and argument values to the procedur ..."
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Cited by 15 (1 self)
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In this thesis, the refinement calculus is extended to support a variety of objectoriented programming styles. The late binding of procedure calls in objectoriented languages is modelled by defining an objectoriented system to be a function from procedure names and argument values to the procedures that are invoked by late binding. The first model allows multiple dispatch late binding, in the style of CLOS. This model is then specialised to the single dispatch case, giving a model that associates types with objects, which is similar to existing class based objectoriented languages. Both models are then restricted so that they support modular reasoning. The concept of modular reasoning has been defined informally in the literature, both for nonobjectoriented systems and for objectoriented systems. This thesis gives the first formal definition of modular reasoning for objectoriented languages. Intuitively, the definition seems to capture the minimum possible requirements necessa...
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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Cited by 8 (5 self)
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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...
Inner Models And Large Cardinals
 Bull. Symbolic Logic
, 1995
"... this paper, we sketch the development of two important themes of modern set theory, ..."
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Cited by 5 (0 self)
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this paper, we sketch the development of two important themes of modern set theory,
Interuniversal Teichmüller Theory IV: Logvolume Computations and Settheoretic Foundations, preprint
"... The present paper forms the fourth and final paper in a series of papers concerning “interuniversal Teichmüller theory”. In the first three papers of the series, we introduced and studied the theory surrounding the logthetalattice, a highly noncommutative twodimensional diagram of “miniature mod ..."
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Cited by 4 (4 self)
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The present paper forms the fourth and final paper in a series of papers concerning “interuniversal Teichmüller theory”. In the first three papers of the series, we introduced and studied the theory surrounding the logthetalattice, a highly noncommutative twodimensional diagram of “miniature models of conventional scheme theory”, called Θ ±ellNFHodge theaters, that were associated, in the first paper of the series, to certain data, called initial Θdata. This data includes an elliptic curve EF over a number field F, together with a prime number l ≥ 5. Consideration of various properties of the logthetalattice led naturally to the establishment, in the third paper of the series, of multiradial algorithms for constructing “splitting monoids of LGPmonoids”. Here, we recall that “multiradial algorithms ” are algorithms that make sense from the point of view of an “alien arithmetic holomorphic structure”, i.e., the ring/scheme structure of a Θ ±ell NFHodge theater related to a given Θ ±ell NFHodge theater by means of a nonring/schemetheoretic horizontal arrow of the logthetalattice. In the present paper, estimates arising from these multiradial algorithms for splitting
The reflection theorem: A study in metatheoretic reasoning
 In Voronkov [19
, 2002
"... Abstract. The reflection theorem has been proved using Isabelle/ZF. This theorem cannot be expressed in ZF, and its proof requires reasoning at the metalevel. There is a particularly elegant proof that reduces the metalevel reasoning to a single induction over formulas. Each case of the induction ..."
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Cited by 4 (2 self)
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Abstract. The reflection theorem has been proved using Isabelle/ZF. This theorem cannot be expressed in ZF, and its proof requires reasoning at the metalevel. There is a particularly elegant proof that reduces the metalevel reasoning to a single induction over formulas. Each case of the induction has been proved with Isabelle/ZF, whose builtin tools can prove specific instances of the reflection theorem upon demand. 1
Reflections on Skolem's Paradox
"... In 1922, Thoraf Skolem published a paper titled "Some remarks on Axiomatized Set Theory". The paper presents a new proof of... This dissertation focuses almost exclusively on the first half of this project  i.e., the half which tries to expose an initial tension between Cantor's theorem and the Lö ..."
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Cited by 4 (2 self)
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In 1922, Thoraf Skolem published a paper titled "Some remarks on Axiomatized Set Theory". The paper presents a new proof of... This dissertation focuses almost exclusively on the first half of this project  i.e., the half which tries to expose an initial tension between Cantor's theorem and the LöwenheimSkolem theorem. I argue that, even on quite naive understandings of set theory and model theory, there is no such tension. Hence, Skolem's Paradox is not a genuine paradox, and there is very little reason to worry about (or even to investigate) the more extreme consequences that are supposed to follow from this paradox. The heart of my...
An Ordinal Analysis of Stability
 ARCHIVE FOR MATHEMATICAL LOGIC
, 2005
"... This paper is the rst in a series of three which culminates in an ordinal analysis of 1 2 comprehension. On the settheoretic side 1 2 comprehension corresponds to KripkePlatek set theory, KP, plus 1 separation. The strength of the latter theory is encapsulated in the fact that it proves th ..."
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Cited by 3 (1 self)
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This paper is the rst in a series of three which culminates in an ordinal analysis of 1 2 comprehension. On the settheoretic side 1 2 comprehension corresponds to KripkePlatek set theory, KP, plus 1 separation. The strength of the latter theory is encapsulated in the fact that it proves the existence of ordinals such that, for all > , is stable, i.e. L is a 1 elementary substructure of L . The objective of this paper is to give an ordinal analysis of not too complicated stability relations as experience has shown that the understanding of the ordinal analysis of 1 2 comprehension is greatly facilated by explicating certain simpler cases rst. This paper introduces an ordinal representation system based on indescribable cardinals which is then employed for determining an upper bound for the proof{ theoretic strength of the theory KPi + 8 9 is + stable, where KPi is KP augmented by the axiom saying that every set is contained in an admissible set.
The Higher Infinite in Proof Theory
 Logic Colloquium '95. Lecture Notes in Logic
, 1995
"... this paper. The exposition here diverges from the presentation given at the conference in two regards. Firstly, the talk began with a broad introduction, explaining the current rationale and goals of ordinaltheoretic proof theory, which take the place of the original Hilbert Program. Since this par ..."
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Cited by 2 (1 self)
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this paper. The exposition here diverges from the presentation given at the conference in two regards. Firstly, the talk began with a broad introduction, explaining the current rationale and goals of ordinaltheoretic proof theory, which take the place of the original Hilbert Program. Since this part of the talk is now incorporated in the first two sections of the BSLpaper [48] there is no point in reproducing it here. Secondly, we shall omit those parts of the talk concerned with infinitary proof systems of ramified set theory as they can also be found in [48] and even more detailed in [45]. Thirdly, thanks to the aforementioned omissions, the advantage of present paper over the talk is to allow for a much more detailed account of the actual information furnished by ordinal analyses and the role of large cardinal hypotheses in devising ordinal representation systems. 2 Observations on ordinal analyses