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Aspects of predicative algebraic set theory I: Exact Completion
 Ann. Pure Appl. Logic
"... This is the first in a series of three papers on Algebraic Set Theory. Its main purpose is to lay the necessary groundwork for the next two parts, one on ..."
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This is the first in a series of three papers on Algebraic Set Theory. Its main purpose is to lay the necessary groundwork for the next two parts, one on
Elementary Axioms for Categories of Classes (Extended Abstract)
 In Proceedings of 14th Annual Symposium on Logic in Computer Science
, 1999
"... ) Dedicated to Ana Alex K. Simpson LFCS, Division of Informatics, University of Edinburgh, JCMB, King's Buildings, Edinburgh, EH9 3JZ Alex.Simpson@dcs.ed.ac.uk Abstract We axiomatize a notion of "classic structure" on a regular category, isolating the essential properties of the category of cla ..."
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) Dedicated to Ana Alex K. Simpson LFCS, Division of Informatics, University of Edinburgh, JCMB, King's Buildings, Edinburgh, EH9 3JZ Alex.Simpson@dcs.ed.ac.uk Abstract We axiomatize a notion of "classic structure" on a regular category, isolating the essential properties of the category of classes together with its full subcategory of sets. Like the axioms for a topos, our axiomatization is very simple, but has powerful consequences. In particular, we show that our axiomatized categories provide a sound and complete class of models for Intuitionistic ZermeloFraenkel set theory. 1. Introduction Almost thirty years on, Lawvere and Tierney's axiomatization of (elementary) toposes [10] remains a milestone in the development of contemporary mathematics. Nowadays, toposes are defined by axioms of exceptional simplicity, which nonetheless entail an astonishing richness of categorical structure [12]. Moreover, toposes are ubiquitous in mathematics. In particular, they have arisen in ge...
Realizability for constructive ZermeloFraenkel set theory
 STOLTENBERGHANSEN (EDS.): PROCEEDINGS OF THE LOGIC COLLOQUIUM 2003
, 2004
"... Constructive ZermeloFraenkel Set Theory, CZF, has emerged as a standard reference theory that relates to constructive predicative mathematics as ZFC relates to classical Cantorian mathematics. A hallmark of this theory is that it possesses a typetheoretic model. Aczel showed that it has a formulae ..."
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Constructive ZermeloFraenkel Set Theory, CZF, has emerged as a standard reference theory that relates to constructive predicative mathematics as ZFC relates to classical Cantorian mathematics. A hallmark of this theory is that it possesses a typetheoretic model. Aczel showed that it has a formulaeastypes interpretation in MartinLöf’s intuitionist theory of types [14, 15]. This paper, though, is concerned with a rather different interpretation. It is shown that Kleene realizability provides a selfvalidating semantics for CZF, viz. this notion of realizability can be formalized in CZF and demonstrably in CZF it can be verified that every theorem of CZF is realized. This semantics, then, is put to use in establishing several equiconsistency results. Specifically, augmenting CZF by wellknown principles germane to Russian constructivism and Brouwer’s intuitionism turns out to engender theories of equal prooftheoretic strength with the same stock of provably recursive functions.
Cut elimination for Zermelo’s set theory
, 2006
"... We show how to express intuitionistic Zermelo set theory in deduction modulo (i.e. by replacing its axioms by rewrite rules) in such a way that the corresponding notion of proof enjoys the normalization property. To do so, we first rephrase set theory as a theory of pointed graphs (following a para ..."
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We show how to express intuitionistic Zermelo set theory in deduction modulo (i.e. by replacing its axioms by rewrite rules) in such a way that the corresponding notion of proof enjoys the normalization property. To do so, we first rephrase set theory as a theory of pointed graphs (following a paradigm due to P. Aczel) by interpreting settheoretic equality as bisimilarity, and show that in this setting, Zermelo’s axioms can be decomposed into graphtheoretic primitives that can be turned into rewrite rules. We then show that the theory we obtain in deduction modulo is a conservative extension of (a minor extension of) Zermelo set theory. Finally, we prove the normalization of the intuitionistic fragment of the theory.
A Strongly Normalising CurryHoward Correspondence for IZF Set Theory
"... Abstract. We propose a method for realising the proofs of Intuitionistic ZermeloFraenkel set theory (IZF) by strongly normalising λterms. This method relies on the introduction of a Currystyle type theory extended with specific subtyping principles, which is then used as a lowlevel language to i ..."
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Abstract. We propose a method for realising the proofs of Intuitionistic ZermeloFraenkel set theory (IZF) by strongly normalising λterms. This method relies on the introduction of a Currystyle type theory extended with specific subtyping principles, which is then used as a lowlevel language to interpret IZF via a representation of sets as pointed graphs inspired by Aczel’s hyperset theory. As a consequence, we refine a classical result of Myhill and Friedman by showing how a strongly normalising λterm that computes a function of type N → N can be extracted from the proof of its existence in IZF. 1
An Interpretation of Kleene's Slash in Type Theory
 Informal Proceedings of the Second Workshop on Logical Frameworks, pages 337342. Esprit Basic Research Action
, 1993
"... Kleene introduced the notion of slash to investigate the disjunction and existence properties under implication for intuitionistic arithmetic. In this paper Kleene's slash is translated to type theory. Besides translations of Kleene's results, the main application of the slash in type theory is that ..."
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Kleene introduced the notion of slash to investigate the disjunction and existence properties under implication for intuitionistic arithmetic. In this paper Kleene's slash is translated to type theory. Besides translations of Kleene's results, the main application of the slash in type theory is that conditions are given for a typable term, containing free variables, to have a normal form beginning with a constructor. 1 Introduction The disjunction and existence properties, that is, ` AB implies ` A or ` B and ` 9xA(x) implies ` A(t) for some term t , respectively, were first proved for intuitionistic arithmetic by Kleene [9] using a modification of recursive realizability. Harrop [8] extended Kleene's result by also considering derivations depending on assumptions. Harrop proved C ` A B implies C ` A or C ` B (ED) C ` 9xA(x) implies C ` A(t) for some term t (EE) where C is a closed formula not containing any strictly positive occurrences of and 9 ; such a formula is called a Har...
Rules and Arithmetics
, 1998
"... This paper is concerned with the `logical structure' of arithmetical theories. We survey results concerning logics and admissible rules of constructive arithmetical theories. We prove a new theorem: the admissible propositional rules of Heyting Arithmetic are the same as the admissible propositional ..."
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This paper is concerned with the `logical structure' of arithmetical theories. We survey results concerning logics and admissible rules of constructive arithmetical theories. We prove a new theorem: the admissible propositional rules of Heyting Arithmetic are the same as the admissible propositional rules of Intuitionistic Propositional Logic. We provide some further insights concerning predicate logical admissible rules for arithmetical theories. Key words: Intuitionistic Logic, Heyting Arithmetic, Kripke models, admissible rules MSC codes: Primary: 03F25, 03F30, Secondary: 0302, 03B20, 03F50, 03F40 Contents 1 Introduction 3 2 Theories and Logics 3 2.1 Propositional Logics of Theories . . . . . . . . . . . . . . . . . . 4 2.2 Predicate Logics of Theories . . . . . . . . . . . . . . . . . . . . . 5 2.3 A Brief History of de Jongh's Theorem . . . . . . . . . . . . . . . 7 2.4 Markov's Principle and Church's Thesis . . . . . . . . . . . . . . 9 2.5 Exactness and Extension . . . . ...
CZF has the disjunction and numerical existence property. Available from the author’s web page www.amsta.leeds.ac.uk/Pure/staff/rathjen/preprints.html
, 2004
"... This paper proves that the disjunction property, the numerical existence property and Church’s rule hold true for Constructive ZermeloFraenkel Set Theory, CZF, and also for the theory CZF augmented by the Regular Extension Axiom. As to the proof technique, it features a selfvalidating semantics fo ..."
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This paper proves that the disjunction property, the numerical existence property and Church’s rule hold true for Constructive ZermeloFraenkel Set Theory, CZF, and also for the theory CZF augmented by the Regular Extension Axiom. As to the proof technique, it features a selfvalidating semantics for CZF that combines extensional Kleene realizability and truth. MSC:03F50, 03F35
Realizability, Set Theory and Term Extraction
"... Applicative Structure : : : : : : : : : : : : : : : : : : : : : : : : : : : : 16 3.2 Realizability : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 17 3.3 Soundness for IZF : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 20 3.4 Term Extraction : : : ..."
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Applicative Structure : : : : : : : : : : : : : : : : : : : : : : : : : : : : 16 3.2 Realizability : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 17 3.3 Soundness for IZF : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 20 3.4 Term Extraction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 21 3.5 Strong Intuitionistic Counterexamples : : : : : : : : : : : : : : : : : : : : : : : : 24 3.6 Some famous "nonstandard" consistency results : : : : : : : : : : : : : : : : : : : 25 4 Forcing in Constructive Set Theory (Unramified) 26 4.1 Kripke models over V (K) : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 26 4.2 Soundness of IZF Axioms : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 28 4.3 Examples : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 30 5 \Omega sets,Categories and Sheaves 34 5.1 From Kripke Models to cHa's : : : : : : : : : : : : : : : :...