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101Compactness under constructive scrutinyMathematical Logic Quarterly 50 (6): 540-550. 2004.How are the various classically equivalent definitions of compactness for metric spaces constructively interrelated? This question is addressed with Bishop-style constructive mathematics as the basic system – that is, the underlying logic is the intuitionistic one enriched with the principle of dependent choices. Besides surveying today's knowledge, the consequences and equivalents of several sequential notions of compactness are investigated. For instance, we establish the perhaps unexpected co…Read more
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147The Fan Theorem and Unique Existence of MaximaJournal of Symbolic Logic 71 (2). 2006.The existence and uniqueness of a maximum point for a continuous real—valued function on a metric space are investigated constructively. In particular, it is shown, in the spirit of reverse mathematics, that a natural unique existence theorem is equivalent to the fan theorem
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105Strong continuity implies uniform sequential continuityArchive for Mathematical Logic 44 (7): 887-895. 2005.Uniform sequential continuity, a property classically equivalent to sequential continuity on compact sets, is shown, constructively, to be a consequence of strong continuity on a metric space. It is then shown that in the case of a separable metric space, uniform sequential continuity implies strong continuity if and only if one adopts a certain boundedness principle that, although valid in the classical, recursive and intuitionistic setting, is independent of Heyting arithmetic.
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84On the contrapositive of countable choiceArchive for Mathematical Logic 50 (1-2): 137-143. 2011.We show that in elementary analysis (EL) the contrapositive of countable choice is equivalent to double negation elimination for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Sigma_{2}^{0}}$$\end{document}-formulas. By also proving a recursive adaptation of this equivalence in Heyting arithmetic (HA), we give an …Read more
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54The Kripke schema in metric topologyMathematical Logic Quarterly 58 (6): 498-501. 2012.A form of Kripke's schema turns out to be equivalent to each of the following two statements from metric topology: every open subspace of a separable metric space is separable; every open subset of a separable metric space is a countable union of open balls. Thus Kripke's schema serves as a point of reference for classifying theorems of classical mathematics within Bishop-style constructive reverse mathematics
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34A direct proof of Wiener's theoremIn S. Barry Cooper (ed.), How the World Computes, . pp. 293--302. 2012.
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81Formal Zariski topology: positivity and pointsAnnals of Pure and Applied Logic 137 (1-3): 317-359. 2006.The topic of this article is the formal topology abstracted from the Zariski spectrum of a commutative ring. After recollecting the fundamental concepts of a basic open and a covering relation, we study some candidates for positivity. In particular, we present a coinductively generated positivity relation. We further show that, constructively, the formal Zariski topology cannot have enough points
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University of LeedsRegular Faculty
Leeds, West Yorkshire, United Kingdom of Great Britain and Northern Ireland
Areas of Interest
| Logic and Philosophy of Logic |
| Medieval and Renaissance Philosophy |