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A BURGESSIAN CRITIQUE OF NOMINALISTIC TENDENCIES IN CONTEMPORARY MATHEMATICS AND ITS HISTORIOGRAPHY
, 2012
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Apartness spaces as framework for constructive topology
 Ann. Pure Appl. Logic
, 2003
"... An axiomatic development of the theory of apartness and nearness of a point and a set is introduced as a framework for constructive topology. Various notions of continuity of mappings between apartness spaces are compared; the constructive independence of one of the axioms from the others is demonst ..."
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Cited by 8 (2 self)
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An axiomatic development of the theory of apartness and nearness of a point and a set is introduced as a framework for constructive topology. Various notions of continuity of mappings between apartness spaces are compared; the constructive independence of one of the axioms from the others is demonstrated; and the product apartness structure is defined and analysed.
MEANING IN CLASSICAL MATHEMATICS: IS IT AT ODDS WITH INTUITIONISM?
, 2011
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Compactness under constructive scrutiny
, 2008
"... The aim of this thesis is to understand the constructive scope of compactness. We show that it is possible to define, constructively, a meaningful notion of compactness in a more general setting than the uniform/metric space one. Furthermore, we show that it is not possible to define compactness co ..."
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The aim of this thesis is to understand the constructive scope of compactness. We show that it is possible to define, constructively, a meaningful notion of compactness in a more general setting than the uniform/metric space one. Furthermore, we show that it is not possible to define compactness constructively in a topological space. We investigate exactly what principles are necessary and sufficient to prove classically true theorems about compactness, as well as their antitheses. We develop beginnings of a constructive theory of differentiable manifolds. i Acknowledgments People to thank fall into the four categories: family, friends, teachers and colleagues—most fall into more than one. This thesis is dedicated to all of them; I am sure that without their contributions this thesis would be emptier. Naturally, D.S. Bridges deserves the biggest share of thanks for his (uniformly)
A Constructive Theory of PointSet Nearness
 in Proceedings of Topology in Computer Science: Constructivity; Asymmetry and Partiality; Digitization, Seminar in Dagstuhl, Germany, 4–9 June 2000; Springer Lecture Notes in Computer Science
, 2001
"... An axiomatic constructive development of the theory of nearness and apartness of a point and a set is introduced as a setting for topology. ..."
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Cited by 3 (1 self)
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An axiomatic constructive development of the theory of nearness and apartness of a point and a set is introduced as a setting for topology.
Real Numbers: From Computable to Random
, 2000
"... A real is computable if it is the limit of a computable, increasing, computably converging sequence of rationals. Omitting the restriction that the sequence converges computably we arrive at the notion of computably enumerable (c.e.) real, that is, the limit of a computable, increasing, converging s ..."
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A real is computable if it is the limit of a computable, increasing, computably converging sequence of rationals. Omitting the restriction that the sequence converges computably we arrive at the notion of computably enumerable (c.e.) real, that is, the limit of a computable, increasing, converging sequence of rationals. A real is random if its binary expansion is a random sequence (equivalently, if its expansion in base b ≥ 2 is random). The aim of this paper is to review some recent results on computable, c.e. and random reals. In particular, we will present a complete characterization of the class of c.e. and random reals in terms of halting probabilities of universal Chaitin machines, and we will show that every c.e. and random real is the halting probability of some Solovay machine, that is, a universal Chaitin machine for which ZFC (if sound) cannot determine more than its initial block of 1 bits. A few open problems will be also discussed. 1 Notation and Background We will use notation that is standard in computability theory and algorithmic information theory; we will assume familiarity with Turing machine computations, computable and computably enumerable (c.e.) sets (see, for example, Soare [48] or Odifreddi [40]) and elementary algorithmic information theory (see,
Randomness Everywhere: Computably Enumerable Reals and Incompleteness
, 2000
"... A real is computable if it is the limit of a computable, increasing, computably converging sequence of rationals. Omitting the restriction that the sequence converges computably we arrive at the notion of computably enumerable (c.e.) real, that is, the limit of a computable, increasing, converging s ..."
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A real is computable if it is the limit of a computable, increasing, computably converging sequence of rationals. Omitting the restriction that the sequence converges computably we arrive at the notion of computably enumerable (c.e.) real, that is, the limit of a computable, increasing, converging sequence of rationals. A real is random if its binary expansion is a random sequence. The aim of these lectures is to review some recent results on computable, c.e. and random reals. In particular, we will present a complete characterization of the class of c.e. and random reals in terms of halting probabilities of universal Chaitin machines, and we will show that every c.e. and random real is the halting probability of some Solovay machine, that is, a universal Chaitin machine for which ZFC (if sound) cannot determine more than its initial block of 1 bits. A few open problems will be also discussed.
Suprema in ordered vector spaces: a constructive approach
"... Ordered vector spaces are examined from the point of view of Bishop’s constructive mathematics, which can be viewed as the constructive core of mathematics. Two different (but classically equivalent) notions of supremum are investigated in order to illustrate some features of constructive mathematic ..."
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Ordered vector spaces are examined from the point of view of Bishop’s constructive mathematics, which can be viewed as the constructive core of mathematics. Two different (but classically equivalent) notions of supremum are investigated in order to illustrate some features of constructive mathematics. By using appropriate definitions of the partial order set, supremum, and ordered vector space, one can prove constructively the usual properties of suprema of subsets of an ordered vector space. Furthermore, the paper provides constructive proofs of the usual properties of the modulus of a vector, proofs that avoid the dichotomy principle, a direct consequence of the law of excluded middle, a law of the classical logic which is viewed as the main source of nonconstructivism.