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Aspects of predicative algebraic set theory II: Realizability. Accepted for publication in Theoretical Computer Science
 In Logic Colloquim 2006, Lecture Notes in Logic
, 2009
"... This is the third in a series of papers on algebraic set theory, the aim of which is to develop a categorical semantics for constructive set theories, including predicative ones, based on the notion of a “predicative category with small maps”. 1 In the first paper in this series [8] we discussed how ..."
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This is the third in a series of papers on algebraic set theory, the aim of which is to develop a categorical semantics for constructive set theories, including predicative ones, based on the notion of a “predicative category with small maps”. 1 In the first paper in this series [8] we discussed how these predicative categories
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
From Sets to Types to Categories to Sets
, 2008
"... Three different styles of foundations of mathematics are now commonplace: set theory, type theory, and category theory. How do they relate, and how do they differ? What advantages and disadvantages does each one have over the others? We pursue these questions by considering interpretations of each s ..."
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Three different styles of foundations of mathematics are now commonplace: set theory, type theory, and category theory. How do they relate, and how do they differ? What advantages and disadvantages does each one have over the others? We pursue these questions by considering interpretations of each system into the others and examining the preservation and loss of mathematical content thereby. In order to stay focused on the “big picture”, we merely sketch the overall form of each construction, referring to the literature for details. Each of the three steps considered below is based on more recent logical research than the preceding one. The first step from sets to types is essentially the familiar idea of set theoretic semantics for a syntactic system, i.e. giving a model; we take a brief glance at this step from the current point of view, mainly just to fix ideas and notation. The second step from types to categories is known to categorical logicians as the construction of a “syntactic category”; we give some specifics for the benefit of the reader who is not familiar with it. The
Published In From Sets to Types to Categories to Sets∗
"... Three different styles of foundations of mathematics are now commonplace: set theory, type theory, and category theory. How do they relate, and how do they differ? What advantages and disadvantages does each one have over the others? We pursue these questions by considering interpretations of each s ..."
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Three different styles of foundations of mathematics are now commonplace: set theory, type theory, and category theory. How do they relate, and how do they differ? What advantages and disadvantages does each one have over the others? We pursue these questions by considering interpretations of each system into the others and examining the preservation and loss of mathematical content thereby. In order to stay focused on the “big picture”, we merely sketch the overall form of each construction, referring to the literature for details. Each of the three steps considered below is based on more recent logical research than the preceding one. The first step from sets to types is essentially the familiar idea of set theoretic semantics for a syntactic system, i.e. giving a model; we take a brief glance at this step from the current point of view, mainly just to fix ideas and notation. The second step from types to categories is known to categorical logicians as the construction of a “syntactic category”; we give some specifics for the benefit of the reader who is not familiar with it. The third step from categories to sets is based on quite recent work, but captures in a precise way an intuition from the early days of foundational studies. With these pieces in place, we can then draw some conclusions regarding the differences between the three schemes, and their relative merits. In particular, it is possible to state more precisely why the methods of category theory are more appropriate to philosophical structuralism. 1 Sets
Basics of Intensionalized Data: Presets, Sets, and Nominats
 COMPUTER SCIENCE JOURNAL OF MOLDOVA, VOL.20, NO.3(60), 2012
, 2012
"... In the paper we consider intensional aspects of the notion of data. We advocate an idea that traditional settheoretic platform should be enhanced with new data structures having explicit intensional component. Among such data we distinguish the notions of preset and nominat. Intuitively, presets ma ..."
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In the paper we consider intensional aspects of the notion of data. We advocate an idea that traditional settheoretic platform should be enhanced with new data structures having explicit intensional component. Among such data we distinguish the notions of preset and nominat. Intuitively, presets may be considered as collections of “black boxes”, nominats may be considered as collections of “grey boxes ” in which “white boxes” are names and “black boxes ” are their values, while sets may be treated as collections of “white boxes”. We describe intensions and properties of the introduced notions. We define operations over such data as functions computable in a special intensionalized sense.
Category theory as an autonomous foundation∗
"... Does category theory provide a foundation for mathematics that is autonomous with respect to the orthodox foundation in a set theory such as ZFC? We distinguish three types of autonomy: logical, conceptual, and justificatory. Focusing on a categorical theory of sets, we argue that a strong case can ..."
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Does category theory provide a foundation for mathematics that is autonomous with respect to the orthodox foundation in a set theory such as ZFC? We distinguish three types of autonomy: logical, conceptual, and justificatory. Focusing on a categorical theory of sets, we argue that a strong case can be made for its logical and conceptual autonomy. Its justificatory autonomy turns on whether the objects of a foundation for mathematics should be specified only up to isomorphism, as is customary in other branches of contemporary mathematics. If such a specification suffices, then a categorytheoretical approach will be highly appropriate. But if sets have a richer ‘nature ’ than is preserved under isomorphism, then such an approach will be inadequate. A number of philosophers of mathematics have recently debated the claim that category theory provides a foundation for mathematics that is autonomous with respect to the orthodox foundation in set theory ([27], [10], [30], [4], [14], [34], [1]). The debate has yielded progress: after some initial confusion, the particular theories from within category theory that are proposed as foundations have been identified precisely, and in some cases the autonomy of these theories with respect to the orthodox foundation has been defended—at least for one sort of autonomy. However, there are other sorts of autonomy that have not been considered in much detail. We wish to introduce a distinction between three types of autonomy, which we call logical autonomy, conceptual autonomy, and justificatory autonomy. The debate so far has been concerned almost exclusively with the first sort of autonomy. Yet all three are required before a foundation can claim genuine independence from the settheoretic orthodoxy.