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28
Safety Signatures for Firstorder Languages and Their Applications
 In FirstOrder Logic Revisited (Hendricks et all,, eds.), 3758, Logos Verlag
, 2004
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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
Algebraic Set Theory and the Effective Topos
, 2005
"... Following the book Algebraic Set Theory from Andre Joyal and Ieke Moerdijk [8], we give a characterization of the initial ZFalgebra, for Heyting pretoposes equipped with a class of small maps. Then, an application is considered (the effective topos) to show how to recover an already known model (Mc ..."
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Following the book Algebraic Set Theory from Andre Joyal and Ieke Moerdijk [8], we give a characterization of the initial ZFalgebra, for Heyting pretoposes equipped with a class of small maps. Then, an application is considered (the effective topos) to show how to recover an already known model (McCarty [9]).
LAWVERETIERNEY SHEAVES IN ALGEBRAIC SET THEORY
"... 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.
Universes in Toposes
, 2004
"... Abstract. We discuss a notion of universe in toposes which from a logical point of view gives rise to an extension of Higher Order Intuitionistic Arithmetic (HAH) that allows one to construct families of types in such a universe by structural recursion and to quantify over such families. Further, we ..."
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Abstract. We discuss a notion of universe in toposes which from a logical point of view gives rise to an extension of Higher Order Intuitionistic Arithmetic (HAH) that allows one to construct families of types in such a universe by structural recursion and to quantify over such families. Further, we show that (hierarchies of) such universes do exist in all sheaf and realizability toposes but neither in the free topos nor in the Vω+ω model of Zermelo set theory. Though universes in Set are necessarily of strongly inaccessible cardinality it remains an open question whether toposes with a universe allow one to construct internal models of Intuitionistic Zermelo Fraenkel set theory (IZF). The background information about toposes and fibred categories as needed for our discussion in this paper can be found e.g. in the fairly accessible sources [MM, Jac, Str2]. 1
An Outline of Algebraic Set Theory
"... This survey article is intended to introduce the reader to the field of Algebraic Set Theory, in which models of set theory of a new and fascinating kind are determined algebraically. The method is quite robust, admitting adjustment in several respects to model different theories including classical ..."
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This survey article is intended to introduce the reader to the field of Algebraic Set Theory, in which models of set theory of a new and fascinating kind are determined algebraically. The method is quite robust, admitting adjustment in several respects to model different theories including classical, intuitionistic, bounded, and predicative ones. Under this scheme some familiar set theoretic properties are related to algebraic ones, like freeness, while others result from logical constraints, like definability. The overall theory is complete in two important respects: conventional elementary set theory axiomatizes algebraic framework itself are also complete with respect to a range of natural models consisting of “ideals ” of sets, suitably defined. Some previous results involving realizability, forcing, and sheaf models are
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.