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Inductive types and exact completion

Benno van den Berg

Annals of Pure and Applied Logic 134 (2-3):95-121 (2005)

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Epistemology Versus Ontology: Essays on the Philosophy and Foundations of Mathematics in Honour of Per Martin-Löf.Peter Dybjer, Sten Lindström, Erik Palmgren & Göran Sundholm (eds.) - 2012 - Dordrecht, Netherland: Springer.details This book brings together philosophers, mathematicians and logicians to penetrate important problems in the philosophy and foundations of mathematics. In philosophy, one has been concerned with the opposition between constructivism and classical mathematics and the different ontological and epistemological views that are reflected in this opposition. The dominant foundational framework for current mathematics is classical logic and set theory with the axiom of choice. This framework is, however, laden with philosophical difficulties. One important alternative foundational programme that is actively pursued (...) Download Export citation Bookmark
Aspects of predicative algebraic set theory I: Exact Completion.Benno van den Berg & Ieke Moerdijk - 2008 - Annals of Pure and Applied Logic 156 (1):123-159.details This is the first in a series of papers on Predicative Algebraic Set Theory, where we lay the necessary groundwork for the subsequent parts, one on realizability [B. van den Berg, I. Moerdijk, Aspects of predicative algebraic set theory II: Realizability, Theoret. Comput. Sci. . Available from: arXiv:0801.2305, 2008], and the other on sheaves [B. van den Berg, I. Moerdijk, Aspects of predicative algebraic set theory III: Sheaf models, 2008 ]. We introduce the notion of a predicative category with small (...) Download Export citation Bookmark 11 citations
Derived rules for predicative set theory: an application of sheaves.Benno van den Berg & Ieke Moerdijk - 2012 - Annals of Pure and Applied Logic 163 (10):1367-1383.details Download Export citation Bookmark 4 citations
Constructivist and structuralist foundations: Bishop’s and Lawvere’s theories of sets.Erik Palmgren - 2012 - Annals of Pure and Applied Logic 163 (10):1384-1399.details Download Export citation Bookmark 9 citations
Non-well-founded trees in categories.Benno van den Berg & Federico De Marchi - 2007 - Annals of Pure and Applied Logic 146 (1):40-59.details Non-well-founded trees are used in mathematics and computer science, for modelling non-well-founded sets, as well as non-terminating processes or infinite data structures. Categorically, they arise as final coalgebras for polynomial endofunctors, which we call M-types. We derive existence results for M-types in locally cartesian closed pretoposes with a natural numbers object, using their internal logic. These are then used to prove stability of such categories with M-types under various topos-theoretic constructions; namely, slicing, formation of coalgebras , and sheaves for an (...) internal site. (shrink) Download Export citation Bookmark 2 citations
Exact completion and constructive theories of sets.Jacopo Emmenegger & Erik Palmgren - 2020 - Journal of Symbolic Logic 85 (2):563-584.details In the present paper we use the theory of exact completions to study categorical properties of small setoids in Martin-Löf type theory and, more generally, of models of the Constructive Elementary Theory of the Category of Sets, in terms of properties of their subcategories of choice objects. Because of these intended applications, we deal with categories that lack equalisers and just have weak ones, but whose objects can be regarded as collections of global elements. In this context, we study the (...) Download Export citation Bookmark

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