Reproduction of all or part of this work is permitted for educational or research use on conditio... more Reproduction of all or part of this work is permitted for educational or research use on condition that this copyright notice is included in any copy. See back inner page for a list of recent BRICS Report Series publications. Copies may be obtained by contacting: BRICS
is permitted for educational or research use on condition that this copyright notice is included ... more is permitted for educational or research use on condition that this copyright notice is included in any copy. See back inner page for a list of recent BRICS Report Series publications. Copies may be obtained by contacting: BRICS
In this paper, we will present a definability theorem for first order logic. This theorem is very... more In this paper, we will present a definability theorem for first order logic. This theorem is very easy to state, and its proof only uses elementary tools. To explain the theorem, let us first observe that if M is a model of a theory T in a language , then, clearly, any definable subset S ⊂ M (i.e., a subset S = {a ∣ M ⊨ φ(a)} defined by some formula φ) is invariant under all automorphisms of M. The same is of course true for subsets of Mn defined by formulas with n free variables.Our theorem states that, if one allows Boolean valued models, the converse holds. More precisely, for any theory T we will construct a Boolean valued model M, in which precisely the T -provable formulas hold, and in which every (Boolean valued) subset which is invariant under all automorphisms of M is definable by a formula .Our presentation is entirely selfcontained, and only requires familiarity with the most elementary properties of model theory. In particular, we have added a first section in which we r...
We prove a topological completeness theorem for infinitary geometric theories with respect to she... more We prove a topological completeness theorem for infinitary geometric theories with respect to sheaf models. The theorem extends a classical result of Makkai and Reyes, stating that any topos with enough points has an open spatial cover. We show that one can achieve in addition that the cover is connected and locally connected. 1
The filter construction, as an endo-functor on the category of small coherent categories, was use... more The filter construction, as an endo-functor on the category of small coherent categories, was used extensively by A. Pitts in a series of papers in the 80's to prove completeness and interpolation results. Later I. Moerdijk and E. Palmgren used the filter construction to construct non-standard models of Heyting arithmetic. In this paper we describe filter construction as a left-adjoint: applied to a left-exact category it is simply the completion of subobject semi-lattices under filtered (and thus all) meets. We study filtered coherent logic which is coherent logic extended to arbitrary meets and the rules that existential quantification and binary disjunction distribute over filtered meets. This logic is sound and complete for interpretations in Pitts' filtered coherent categories, and conservative over coherent logic. Restricting further to first-order logic we show that the minimal models of Heyting arithmetic described by Moerdijk and Palmgren are nothing but generic m...
We construct Boolean-valued term models and describe their universal properties. The underlying s... more We construct Boolean-valued term models and describe their universal properties. The underlying set of these models consists of the terms of the underlying language, while formulae are interpreted in the Lindenbaum-Tarski algebra of open formulae. Subspaces of the Stone space of the Lindenbaum-Tarski algebra of open formulae are shown to be homeomorphic to spaces of enumerated models- models equipped with a quotient map from some fixed large cardinal. The motivation for writing this note was two-fold. First of all we wanted to know the simplest possible construction of a Boolean-valued model for a given theory. It is not that we did not know how to construct a Boolean-valued model. In fact, together with Ieke Moerdijk, the author has presented a simple construction of Boolean-valued models for first-order theories, see [4]. That construction takes a detour by using completeness of first-order logic with respect to models in sets. A set of ’enough ’ models (equipped with some additio...
The performance of data parallel programs often hinges on two key coordination aspects: the compu... more The performance of data parallel programs often hinges on two key coordination aspects: the computational costs of the parallel tasks relative to their management overhead | task granularity ; and the communication costs induced by the distance between tasks and their data | data locality . In data parallel programs both granularity and locality can be improved by clustering, i.e. arranging for parallel tasks to operate on related sub-collections of data.
Abstract. We show how to interpret the language of first-order set theory in an elementary topos ... more Abstract. We show how to interpret the language of first-order set theory in an elementary topos endowed with, as extra structure, a directed structural system of inclusions (dssi). As our main result, we obtain a complete axiomatization of the intuitionistic set theory validated by all such interpretations. Since every elementary topos is equivalent to one carrying a dssi, we thus obtain a first-order set theory whose associated categories of sets are exactly the elementary toposes. In addition, we show that the full axiom of Separation is validated whenever the dssi is superdirected. This gives a uniform explanation for the known facts that cocomplete and realizability toposes provide models for Intuitionistic Zermelo–Fraenkel set theory (IZF). §1. Introduction. The notion of elementary topos abstracts from the structure of the category of sets. The abstraction is sufficiently general that elementary toposes encompass a rich collection of other very different categories, including...
In this paper we will present a de nability theorem for rst order logic This theorem is very easy... more In this paper we will present a de nability theorem for rst order logic This theorem is very easy to state and its proof only uses elementary tools To explain the theorem let us rst observe that if M is a model of a theory T in a language L then clearly any de nable subset S M i e a subset S fa j M j a g de ned by some formula is invariant under all automorphisms of M The same is of course true for subsets of M n de ned by formulas with n free variables Our theorem states that if one allows Boolean valued models the converse holds More precisely for any theory T we will construct a Boolean valued model M in which precisely the T provable formulas hold and in which every Boolean valued subset which is invariant under all automorphisms of M is de nable by a formula of L Our presentation is entirely selfcontained and only requires familiarity with the most elementary properties of model theory In particular we have added a rst section in which we review the basic de nitions concerning ...
We study the type-theoretical analogue of Bernays–G&odel set-theory and its models in categories.... more We study the type-theoretical analogue of Bernays–G&odel set-theory and its models in categories. We introduce the notion of small structure on a category, and if small structure satis5es certain axioms we can think of the underlying category as a category of classes. Our axioms imply the existence of a co-variant powerset monad on the underlying category of classes, which sends a class to the class of its small subclasses. Simple 5xed points of this and related monads are shown to be models of intuitionistic Zermelo–Fraenkel set-theory (IZF). c
The lter construction, as an endo-functor on the category of small coherent categories, was used ... more The lter construction, as an endo-functor on the category of small coherent categories, was used extensively by A. Pitts in a series of papers in the 80's to prove completeness and interpolation results. Later I. Moerdijk and E. Palmgren used the lter construction to construct non-standard models of Heyting arithmetic. In this paper we describe lter construction as a left-adjoint: applied to a left-exact category it is simply the completion of subobject semi-lattices under ltered (and thus all) meets. We study ltered coherent logic which is coherent logic extended to arbitrary meets and the rules that existential quantiication and binary disjunction distribute over ltered meets. This logic is sound and complete for interpretations in Pitts' ltered coherent categories, and conservative over coherent logic. Restricting further to rst-order logic we show that the minimal models of Heyting arithmetic described by Moerdijk and Palmgren are nothing but generic models of Heyting ar...
These notes were supposed to give more detailed information about the relationship between regula... more These notes were supposed to give more detailed information about the relationship between regular categories and regular logic than is contained in Jaap van Oosten's script on category theory (BRICS Lectures Series LS-95-1). Regular logic is there called coherent logic. I would like to thank Jaap van Oosten for some comments on these notes.
Reproduction of all or part of this work is permitted for educational or research use on conditio... more Reproduction of all or part of this work is permitted for educational or research use on condition that this copyright notice is included in any copy. See back inner page for a list of recent BRICS Report Series publications. Copies may be obtained by contacting: BRICS
is permitted for educational or research use on condition that this copyright notice is included ... more is permitted for educational or research use on condition that this copyright notice is included in any copy. See back inner page for a list of recent BRICS Report Series publications. Copies may be obtained by contacting: BRICS
In this paper, we will present a definability theorem for first order logic. This theorem is very... more In this paper, we will present a definability theorem for first order logic. This theorem is very easy to state, and its proof only uses elementary tools. To explain the theorem, let us first observe that if M is a model of a theory T in a language , then, clearly, any definable subset S ⊂ M (i.e., a subset S = {a ∣ M ⊨ φ(a)} defined by some formula φ) is invariant under all automorphisms of M. The same is of course true for subsets of Mn defined by formulas with n free variables.Our theorem states that, if one allows Boolean valued models, the converse holds. More precisely, for any theory T we will construct a Boolean valued model M, in which precisely the T -provable formulas hold, and in which every (Boolean valued) subset which is invariant under all automorphisms of M is definable by a formula .Our presentation is entirely selfcontained, and only requires familiarity with the most elementary properties of model theory. In particular, we have added a first section in which we r...
We prove a topological completeness theorem for infinitary geometric theories with respect to she... more We prove a topological completeness theorem for infinitary geometric theories with respect to sheaf models. The theorem extends a classical result of Makkai and Reyes, stating that any topos with enough points has an open spatial cover. We show that one can achieve in addition that the cover is connected and locally connected. 1
The filter construction, as an endo-functor on the category of small coherent categories, was use... more The filter construction, as an endo-functor on the category of small coherent categories, was used extensively by A. Pitts in a series of papers in the 80's to prove completeness and interpolation results. Later I. Moerdijk and E. Palmgren used the filter construction to construct non-standard models of Heyting arithmetic. In this paper we describe filter construction as a left-adjoint: applied to a left-exact category it is simply the completion of subobject semi-lattices under filtered (and thus all) meets. We study filtered coherent logic which is coherent logic extended to arbitrary meets and the rules that existential quantification and binary disjunction distribute over filtered meets. This logic is sound and complete for interpretations in Pitts' filtered coherent categories, and conservative over coherent logic. Restricting further to first-order logic we show that the minimal models of Heyting arithmetic described by Moerdijk and Palmgren are nothing but generic m...
We construct Boolean-valued term models and describe their universal properties. The underlying s... more We construct Boolean-valued term models and describe their universal properties. The underlying set of these models consists of the terms of the underlying language, while formulae are interpreted in the Lindenbaum-Tarski algebra of open formulae. Subspaces of the Stone space of the Lindenbaum-Tarski algebra of open formulae are shown to be homeomorphic to spaces of enumerated models- models equipped with a quotient map from some fixed large cardinal. The motivation for writing this note was two-fold. First of all we wanted to know the simplest possible construction of a Boolean-valued model for a given theory. It is not that we did not know how to construct a Boolean-valued model. In fact, together with Ieke Moerdijk, the author has presented a simple construction of Boolean-valued models for first-order theories, see [4]. That construction takes a detour by using completeness of first-order logic with respect to models in sets. A set of ’enough ’ models (equipped with some additio...
The performance of data parallel programs often hinges on two key coordination aspects: the compu... more The performance of data parallel programs often hinges on two key coordination aspects: the computational costs of the parallel tasks relative to their management overhead | task granularity ; and the communication costs induced by the distance between tasks and their data | data locality . In data parallel programs both granularity and locality can be improved by clustering, i.e. arranging for parallel tasks to operate on related sub-collections of data.
Abstract. We show how to interpret the language of first-order set theory in an elementary topos ... more Abstract. We show how to interpret the language of first-order set theory in an elementary topos endowed with, as extra structure, a directed structural system of inclusions (dssi). As our main result, we obtain a complete axiomatization of the intuitionistic set theory validated by all such interpretations. Since every elementary topos is equivalent to one carrying a dssi, we thus obtain a first-order set theory whose associated categories of sets are exactly the elementary toposes. In addition, we show that the full axiom of Separation is validated whenever the dssi is superdirected. This gives a uniform explanation for the known facts that cocomplete and realizability toposes provide models for Intuitionistic Zermelo–Fraenkel set theory (IZF). §1. Introduction. The notion of elementary topos abstracts from the structure of the category of sets. The abstraction is sufficiently general that elementary toposes encompass a rich collection of other very different categories, including...
In this paper we will present a de nability theorem for rst order logic This theorem is very easy... more In this paper we will present a de nability theorem for rst order logic This theorem is very easy to state and its proof only uses elementary tools To explain the theorem let us rst observe that if M is a model of a theory T in a language L then clearly any de nable subset S M i e a subset S fa j M j a g de ned by some formula is invariant under all automorphisms of M The same is of course true for subsets of M n de ned by formulas with n free variables Our theorem states that if one allows Boolean valued models the converse holds More precisely for any theory T we will construct a Boolean valued model M in which precisely the T provable formulas hold and in which every Boolean valued subset which is invariant under all automorphisms of M is de nable by a formula of L Our presentation is entirely selfcontained and only requires familiarity with the most elementary properties of model theory In particular we have added a rst section in which we review the basic de nitions concerning ...
We study the type-theoretical analogue of Bernays–G&odel set-theory and its models in categories.... more We study the type-theoretical analogue of Bernays–G&odel set-theory and its models in categories. We introduce the notion of small structure on a category, and if small structure satis5es certain axioms we can think of the underlying category as a category of classes. Our axioms imply the existence of a co-variant powerset monad on the underlying category of classes, which sends a class to the class of its small subclasses. Simple 5xed points of this and related monads are shown to be models of intuitionistic Zermelo–Fraenkel set-theory (IZF). c
The lter construction, as an endo-functor on the category of small coherent categories, was used ... more The lter construction, as an endo-functor on the category of small coherent categories, was used extensively by A. Pitts in a series of papers in the 80's to prove completeness and interpolation results. Later I. Moerdijk and E. Palmgren used the lter construction to construct non-standard models of Heyting arithmetic. In this paper we describe lter construction as a left-adjoint: applied to a left-exact category it is simply the completion of subobject semi-lattices under ltered (and thus all) meets. We study ltered coherent logic which is coherent logic extended to arbitrary meets and the rules that existential quantiication and binary disjunction distribute over ltered meets. This logic is sound and complete for interpretations in Pitts' ltered coherent categories, and conservative over coherent logic. Restricting further to rst-order logic we show that the minimal models of Heyting arithmetic described by Moerdijk and Palmgren are nothing but generic models of Heyting ar...
These notes were supposed to give more detailed information about the relationship between regula... more These notes were supposed to give more detailed information about the relationship between regular categories and regular logic than is contained in Jaap van Oosten's script on category theory (BRICS Lectures Series LS-95-1). Regular logic is there called coherent logic. I would like to thank Jaap van Oosten for some comments on these notes.
Uploads
Papers by Carsten Butz