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  1. Logic, Mathematics, Philosophy, Vintage Enthusiasms: Essays in Honour of John L. Bell.David DeVidi, Michael Hallett & Peter Clark (eds.) - 2011 - Dordrecht, Netherland: Springer.
    The volume includes twenty-five research papers presented as gifts to John L. Bell to celebrate his 60th birthday by colleagues, former students, friends and admirers. Like Bell’s own work, the contributions cross boundaries into several inter-related fields. The contributions are new work by highly respected figures, several of whom are among the key figures in their fields. Some examples: in foundations of maths and logic ; analytical philosophy, philosophy of science, philosophy of mathematics and decision theory and foundations of economics. (...)
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  • A coverage construction of the reals and the irrationals.Harold Simmons - 2007 - Annals of Pure and Applied Logic 145 (2):176-203.
    I modify the standard coverage construction of the reals to obtain the irrationals. However, this causes a jump in ordinal complexity from ω+1 to Ω.
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  • Why Topology in the Minimalist Foundation Must be Pointfree.Maria Emilia Maietti & Giovanni Sambin - 2013 - Logic and Logical Philosophy 22 (2):167-199.
    We give arguments explaining why, when adopting a minimalist approach to constructive mathematics as that formalized in our two-level minimalist foundation, the choice for a pointfree approach to topology is not just a matter of convenience or mathematical elegance, but becomes compulsory. The main reason is that in our foundation real numbers, either as Dedekind cuts or as Cauchy sequences, do not form a set.
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  • Heyting-valued interpretations for constructive set theory.Nicola Gambino - 2006 - Annals of Pure and Applied Logic 137 (1-3):164-188.
    We define and investigate Heyting-valued interpretations for Constructive Zermelo–Frankel set theory . These interpretations provide models for CZF that are analogous to Boolean-valued models for ZF and to Heyting-valued models for IZF. Heyting-valued interpretations are defined here using set-generated frames and formal topologies. As applications of Heyting-valued interpretations, we present a relative consistency result and an independence proof.
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  • Regular opens in constructive topology and a representation theorem for overlap algebras.Francesco Ciraulo - 2013 - Annals of Pure and Applied Logic 164 (4):421-436.
    Giovanni Sambin has recently introduced the notion of an overlap algebra in order to give a constructive counterpart to a complete Boolean algebra. We propose a new notion of regular open subset within the framework of intuitionistic, predicative topology and we use it to give a representation theorem for overlap algebras. In particular we show that there exists a duality between the category of set-based overlap algebras and a particular category of topologies in which all open subsets are regular.
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  • Constructive version of Boolean algebra.F. Ciraulo, M. E. Maietti & P. Toto - 2013 - Logic Journal of the IGPL 21 (1):44-62.
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  • Embedding locales and formal topologies into positive topologies.Francesco Ciraulo & Giovanni Sambin - 2018 - Archive for Mathematical Logic 57 (7-8):755-768.
    A positive topology is a set equipped with two particular relations between elements and subsets of that set: a convergent cover relation and a positivity relation. A set equipped with a convergent cover relation is a predicative counterpart of a locale, where the given set plays the role of a set of generators, typically a base, and the cover encodes the relations between generators. A positivity relation enriches the structure of a locale; among other things, it is a tool to (...)
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  • Factorizing the $$\mathbf {Top}$$ Top – $$\mathbf {Loc}$$ Loc adjunction through positive topologies.Francesco Ciraulo, Tatsuji Kawai & Samuele Maschio - 2021 - Archive for Mathematical Logic 60 (7):967-979.
    We characterize the category of Sambin’s positive topologies as the result of the Grothendieck construction applied to a doctrine over the category Loc of locales. We then construct an adjunction between the category of positive topologies and that of topological spaces Top, and show that the well-known adjunction between Top and Loc factors through the constructed adjunction.
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  • Factorizing the Top\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {Top}$$\end{document}–Loc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {Loc}$$\end{document} adjunction through positive topologies. [REVIEW]Francesco Ciraulo, Tatsuji Kawai & Samuele Maschio - 2021 - Archive for Mathematical Logic 60 (7-8):967-979.
    We characterize the category of Sambin’s positive topologies as the result of the Grothendieck construction applied to a doctrine over the category Loc of locales. We then construct an adjunction between the category of positive topologies and that of topological spaces Top, and show that the well-known adjunction between Top and Loc factors through the constructed adjunction.
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  • Maddy On The Multiverse.Claudio Ternullo - 2019 - In Deniz Sarikaya, Deborah Kant & Stefania Centrone (eds.), Reflections on the Foundations of Mathematics. Berlin: Springer Verlag. pp. 43-78.
    Penelope Maddy has recently addressed the set-theoretic multiverse, and expressed reservations on its status and merits ([Maddy, 2017]). The purpose of the paper is to examine her concerns, by using the interpretative framework of set-theoretic naturalism. I first distinguish three main forms of 'multiversism', and then I proceed to analyse Maddy's concerns. Among other things, I take into account salient aspects of multiverse-related mathematics , in particular, research programmes in set theory for which the use of the multiverse seems to (...)
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