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36
Choice principles in constructive and classical set theories
 POHLERS (EDS.): PROCEEDINGS OF THE LOGIC COLLOQUIUM 2002
, 2002
"... The objective of this paper is to assay several forms of the axiom of choice that have been deemed constructive. In addition to their deductive relationships, the paper will be concerned with metamathematical properties effected by these choice principles and also with some of their classical models ..."
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The objective of this paper is to assay several forms of the axiom of choice that have been deemed constructive. In addition to their deductive relationships, the paper will be concerned with metamathematical properties effected by these choice principles and also with some of their classical models.
Models of nonwellfounded sets via an indexed final coalgebra theorem
 J. Symbolic Logic
"... ..."
THE ASSOCIATED SHEAF FUNCTOR THEOREM IN ALGEBRAIC SET THEORY
"... Abstract. We prove a version of the associated sheaf functor theorem in Algebraic Set Theory. The proof is established working within a Heyting pretopos equipped with a system of small maps satisfying the axioms originally introduced by Joyal and Moerdijk. This result improves on the existing develo ..."
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Abstract. We prove a version of the associated sheaf functor theorem in Algebraic Set Theory. The proof is established working within a Heyting pretopos equipped with a system of small maps satisfying the axioms originally introduced by Joyal and Moerdijk. This result improves on the existing developments by avoiding the assumption of additional axioms for small maps and the use of collection sites. 1.
Categorical models of intuitionistic theories of sets and classes
, 2004
"... The thesis consists of three sections, developing models of intuitionistic set theory in suitable categories. First, the categorical framework in which models are constructed is reviewed, and the theory of all such models, called Basic Intuitionistic Set Theory (BIST), is stated; second, we give a n ..."
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The thesis consists of three sections, developing models of intuitionistic set theory in suitable categories. First, the categorical framework in which models are constructed is reviewed, and the theory of all such models, called Basic Intuitionistic Set Theory (BIST), is stated; second, we give a notion of an ideal over a category, with which one can build a model of BIST in which a given topos occurs as the sets; and third, a sheaf model is given of a Basic Intuitionistic Class Theory conservatively extending BIST.
Algebraic models of sets and classes in categories of ideals
 In preparation
, 2006
"... We introduce a new sheaftheoretic construction called the ideal completion of a category and investigate its logical properties. We show that it satisfies the axioms for a category of classes in the sense of Joyal and Moerdijk [17], so that the tools of algebraic set theory can be applied to produc ..."
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We introduce a new sheaftheoretic construction called the ideal completion of a category and investigate its logical properties. We show that it satisfies the axioms for a category of classes in the sense of Joyal and Moerdijk [17], so that the tools of algebraic set theory can be applied to produce models of various elementary set theories. These results are then used to prove the conservativity of different set theories over various classical and constructive type theories. 1
Relating firstorder set theories, toposes and categories of classes
 In preparation
, 2006
"... This paper introduces Basic Intuitionistic Set Theory BIST, and investigates it as a firstorder settheory extending the internal logic of elementary toposes. Given an elementary topos, together with the extra structure of a directed structural system of inclusions (dssi) on the topos, a forcingst ..."
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This paper introduces Basic Intuitionistic Set Theory BIST, and investigates it as a firstorder settheory extending the internal logic of elementary toposes. Given an elementary topos, together with the extra structure of a directed structural system of inclusions (dssi) on the topos, a forcingstyle interpretation of the language of firstorder set theory in the topos is given, which conservatively extends the internal logic of the topos. Since every topos is equivalent to one carrying a dssi, the language of firstorder has a forcing interpretation in every elementary topos. We prove that the set theory BIST+ Coll (where Coll is the strong Collection axiom) is sound and complete relative to forcing interpretations in toposes with natural numbers object (nno). Furthermore, in the case that the structural system of inclusions is superdirected, the full Separation schema is modelled. We show that every cocomplete topos and every realizability topos can be endowed (up to equivalence) with such a superdirected structural system of inclusions. This provides a uniform explanation for why such “realworld ” toposes model Separation. A large part of the paper is devoted to an alternative notion of categorytheoretic model for BIST, which, following the general approach of Joyal and Moerdijk’s Algebraic Set Theory, axiomatizes the structure possessed by categories of classes compatible with ∗Corresponding author. 1Previously, lecturer at HeriotWatt University (2000–2001), and the IT University of
Formalizing abstract algebra in constructive set theory
, 2002
"... Abstract. We present a machinechecked formalization of elementary abstract algebra in constructive set theory. Our formalization uses an approach where we start by specifying the group axioms as a collection of inference rules, defining a logic for groups. Then we can tell whether a given set with ..."
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Abstract. We present a machinechecked formalization of elementary abstract algebra in constructive set theory. Our formalization uses an approach where we start by specifying the group axioms as a collection of inference rules, defining a logic for groups. Then we can tell whether a given set with a binary operation is a group or not, and derive all properties of groups constructively from these inference rules as well as the axioms of the set theory. The formalization of all other concepts in abstract algebra is based on that of the group. We give an example of a formalization of a concrete group, the Klein 4group. 1
Some Mathematical Case Studies in ProofPowerHOL
, 2004
"... TPHOLs conference covers all aspects of theorem proving in higher order logics as well as related topics in theorem proving and verification. There were 42 papers submitted to TPHOLs 2004 in the full research category, each of which was refereed by at least 3 reviewers selected by the program commit ..."
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TPHOLs conference covers all aspects of theorem proving in higher order logics as well as related topics in theorem proving and verification. There were 42 papers submitted to TPHOLs 2004 in the full research category, each of which was refereed by at least 3 reviewers selected by the program committee. Of these submissions, 21 were accepted for presentation at the conference and publication in volume 3223 of Springer’s Lecture Notes in Computer Science series. In keeping with longstanding tradition, TPHOLs 2004 also offered a venue for the presentation of work in progress, where researchers invite discussion by means of a brief introductory talk and then discuss their work at a poster
A general construction of internal sheaves in algebraic set theory. Preliminary version available at [3
"... Abstract. We present a solution to the problem of defining a counterpart in Algebraic Set Theory of the construction of internal sheaves in Topos Theory. Our approach is general in that we consider sheaves as determined by LawvereTierney coverages, rather than by Grothendieck coverages, and assume ..."
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Abstract. We present a solution to the problem of defining a counterpart in Algebraic Set Theory of the construction of internal sheaves in Topos Theory. Our approach is general in that we consider sheaves as determined by LawvereTierney coverages, rather than by Grothendieck coverages, and assume only a weakening of the axioms for small maps originally introduced by Joyal and Moerdijk, thus subsuming the existing topostheoretic results.
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