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16
Recursive coalgebras of finitary functors
 Department of Computer Science, University of Wales Swansea
, 2005
"... Abstract For finitary set functors preserving inverse images several concepts of coalgebras A are proved to be equivalent: (i) A has a homomorphism into the initial algebra, (ii) A is recursive, i.e., A has a unique coalgebratoalgebra morphism into any algebra, and (iii) A is parametrically recurs ..."
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Abstract For finitary set functors preserving inverse images several concepts of coalgebras A are proved to be equivalent: (i) A has a homomorphism into the initial algebra, (ii) A is recursive, i.e., A has a unique coalgebratoalgebra morphism into any algebra, and (iii) A is parametrically recursive. And all these properties mean that the system described by A always halts in finitely many steps. 1
Does category theory provide a framework for mathematical structuralism?
 PHILOSOPHIA MATHEMATICA
, 2003
"... Category theory and topos theory have been seen as providing a structuralist framework for mathematics autonomous vis à vis set theory. It is argued here that these theories require a background logic of relations and substantive assumptions addressing mathematical existence of categories themselves ..."
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Category theory and topos theory have been seen as providing a structuralist framework for mathematics autonomous vis à vis set theory. It is argued here that these theories require a background logic of relations and substantive assumptions addressing mathematical existence of categories themselves. We propose a synthesis of Bell’s “manytopoi” view and modalstructuralism. Surprisingly, a combination of mereology and plural quantification suffices to describe hypothetical large domains, recovering the Grothendieck method of universes. Both topos theory and set theory can be carried out relative to such domains; puzzles about “large categories ” and “proper classes ” are handled in a
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
Corecursive Algebras: A Study of General Structured Corecursion (Extended Abstract)
"... Abstract. We study general structured corecursion, dualizing the work of Osius, Taylor, and others on general structured recursion. We call an algebra of a functor corecursive if it supports general structured corecursion: there is a unique map to it from any coalgebra of the same functor. The conce ..."
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Abstract. We study general structured corecursion, dualizing the work of Osius, Taylor, and others on general structured recursion. We call an algebra of a functor corecursive if it supports general structured corecursion: there is a unique map to it from any coalgebra of the same functor. The concept of antifounded algebra is a statement of the bisimulation principle. We show that it is independent from corecursiveness: Neither condition implies the other. Finally, we call an algebra focusing if its codomain can be reconstructed by iterating structural refinement. This is the strongest condition and implies all the others. 1
Constructivist and Structuralist Foundations: Bishop’s and Lawvere’s Theories of Sets
, 2009
"... Bishop’s informal set theory is briefly discussed and compared to Lawvere’s Elementary Theory of the Category of Sets (ETCS). We then present a constructive and predicative version of ETCS, whose standard model is based on the constructive type theory of MartinLöf. The theory, CETCS, provides a str ..."
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Bishop’s informal set theory is briefly discussed and compared to Lawvere’s Elementary Theory of the Category of Sets (ETCS). We then present a constructive and predicative version of ETCS, whose standard model is based on the constructive type theory of MartinLöf. The theory, CETCS, provides a structuralist foundation for constructive mathematics in the style of Bishop.
RCF 4 Inconsistent Quantification AC ∀∃! ε
, 901
"... We exhibit canonical middleinverse Choice maps within categorical (FreeVariable) Theory of Primitive Recursion as well as in Theory of partial PR maps over Theory of Primitive Recursion with predicate abstraction. Using these choicemaps, defined by µrecursion, we address the consistency problem ..."
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We exhibit canonical middleinverse Choice maps within categorical (FreeVariable) Theory of Primitive Recursion as well as in Theory of partial PR maps over Theory of Primitive Recursion with predicate abstraction. Using these choicemaps, defined by µrecursion, we address the consistency problem for a minimal Quantified extension Q of latter two theories: We prove, that Q’s ∃defined µoperator coincides on PR predicates with that inherited from theory of partial PR maps. We strengthen Theory Q by axiomatically forcing the lexicographical order on its ω ω to become a wellorder: “finite descent”. Resulting theory admits noninfinit PRiterative descent schema (π) which constitutes Cartesian PR Theory πR introduced in RCF2. A suitable Cartesian subSystem of Q + wo(ω ω) above, extension of πR “inside ” Theory Q+wo(ω ω), is shown to admit code selfevaluation: extension of formally partial code evaluation of πR. Appropriate diagonal argument then shows inconsistency of this subSystem and (hence) of its extensions Q + wo(ω ω) and ZF. 1
SET THEORY FOR CATEGORY THEORY
, 810
"... Abstract. Questions of settheoretic size play an essential role in category theory, especially the distinction between sets and proper classes (or small sets and large sets). There are many different ways to formalize this, and which choice is made can have noticeable effects on what categorical co ..."
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Abstract. Questions of settheoretic size play an essential role in category theory, especially the distinction between sets and proper classes (or small sets and large sets). There are many different ways to formalize this, and which choice is made can have noticeable effects on what categorical constructions are permissible. In this expository paper we summarize and compare a number
RCF 4 Inconsistent Quantification
, 2008
"... We exhibit canonical Choice maps within categorical theories of Primitive Recursion, of partially defined PR maps, as well as for classical, quantifier defined PR theories, and show incompatibility of these choice sections in the latter theories, with (iterative) finitedescent property of ω ω, name ..."
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We exhibit canonical Choice maps within categorical theories of Primitive Recursion, of partially defined PR maps, as well as for classical, quantifier defined PR theories, and show incompatibility of these choice sections in the latter theories, with (iterative) finitedescent property of ω ω, namely within a “minimal ” such quantor defined Arithmetic, Q. This is to give inconsistency of ZF, and even of first order set theory 1ZF strengthened by wellorder property of ω ω. The argument is iterative evaluation of PR map codes, which gets epimorphic definedarguments enumeration by above finitedescent property. This enumeration is turned into a retraction by AC, with PR section in Q + = Q + wo(ω ω), and so makes the evaluation a PR map. But the latter is excluded by Ackermann’s result that such (diagonalised) evaluation grows faster than any PR map within any consistent frame.
Only up to isomorphism? Category theory and the . . .
"... 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.