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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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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.
Sequential continuity of linear mappings in constructive mathematics
 J. Universal Computer Science
, 1997
"... Abstract: This paper deals, constructively, with two theorems on the sequential continuity of linear mappings. The classical proofs of these theorems use the boundedness of the linear mappings, which is a constructively stronger property than sequential continuity; and constructively inadmissable ve ..."
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Abstract: This paper deals, constructively, with two theorems on the sequential continuity of linear mappings. The classical proofs of these theorems use the boundedness of the linear mappings, which is a constructively stronger property than sequential continuity; and constructively inadmissable versions of the BanachSteinhaus theorem.
Sequentially Continuous Linear Mappings in Constructive Analysis
, 1996
"... this paper we derive some results about sequentially continuous linear mappings within BISH. These results tend to reinforce our hope that such mappings may turn out to be bounded (continuous) after all. For background material on BISH, see [1], and for information about the relation between BISH, I ..."
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this paper we derive some results about sequentially continuous linear mappings within BISH. These results tend to reinforce our hope that such mappings may turn out to be bounded (continuous) after all. For background material on BISH, see [1], and for information about the relation between BISH, INT, and RUSS, see [2]. 2 Sequential continuity preserves Cauchyness
A constructive look at functions of bounded variation
 Bull. London Math. Soc
"... Functions with bounded variation and with a (total) variation are examined within Bishop’s constructive mathematics. It is shown that the property of having a variation is hereditary downward on compact intervals, and hence that a realvalued function f with a variation on a compact interval can be ..."
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Functions with bounded variation and with a (total) variation are examined within Bishop’s constructive mathematics. It is shown that the property of having a variation is hereditary downward on compact intervals, and hence that a realvalued function f with a variation on a compact interval can be expressed as a dierence of two increasing functions. Moreover, if f is sequentially continuous, then the corresponding variation function, and hence f itself, is uniformly continuous. 1.
Arguments for the Continuity Principle
, 2000
"... Contents 1 The continuity principle 1 2 A phenomenological consideration 8 2.1 An argument for G(raph)WCN . . . . . . . . . . . . . . . . . 8 2.2 Two arguments against WCN . . . . . . . . . . . . . . . . . . 13 3 Other arguments for continuity 15 3.1 Undecidability of equality of choice sequences ..."
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Cited by 5 (1 self)
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Contents 1 The continuity principle 1 2 A phenomenological consideration 8 2.1 An argument for G(raph)WCN . . . . . . . . . . . . . . . . . 8 2.2 Two arguments against WCN . . . . . . . . . . . . . . . . . . 13 3 Other arguments for continuity 15 3.1 Undecidability of equality of choice sequences . . . . . . . . . 15 3.2 Kripke's Schema and full PEM . . . . . . . . . . . . . . . . . 15 3.3 The KLST theorem . . . . . . . . . . . . . . . . . . . . . . . . 16 4 Conclusion 19 1 The continuity principle There are two principles that lend Brouwer's mathematics the extra power beyond arithmetic. Both are presented in Brouwer's writings with little or no argument. One, the principle of bar induction, will not concern us here. The other, the continuity principle for numbers, occurs for the rst time in print in [Brouwer 1918]. It is formulated and immediately applied to show that the set of numerical choice sequences is not enumerable. In fa
On weak Markov's principle
 MLQ MATH. LOG. Q
, 2002
"... We show that the socalled weak Markov's principle (WMP) which states that every pseudopositive real number is positive is underivable in T # :=EHA # +AC. Since T # allows to formalize (at least large parts of) Bishop's constructive mathematics this makes it unlikely that WMP can ..."
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We show that the socalled weak Markov's principle (WMP) which states that every pseudopositive real number is positive is underivable in T # :=EHA # +AC. Since T # allows to formalize (at least large parts of) Bishop's constructive mathematics this makes it unlikely that WMP can be proved within the framework of Bishopstyle mathematics (which has been open for about 20 years). The underivability even holds if the ine#ective schema of full comprehension (in all types) for negated formulas (in particular for #free formulas) is added which allows to derive the law of excluded middle for such formulas.
Infinite sets that satisfy the principle of omniscience in all varieties of constructive mathematics, MartinLöf formalization, in Agda notation, of part of the paper with the same title
 University of Birmingham, UK, http://www.cs.bham.ac.uk/~mhe/papers/ omniscient/AnInfiniteOmniscientSet.html, September 2011. SETS IN CONSTRUCTIVE MATHEMATICS 21
"... Abstract. We show that there are plenty of infinite sets that satisfy the omniscience principle, in a minimalistic setting for constructive mathematics that is compatible with classical mathematics. A first example of an omniscient set is the onepoint compactification of the natural numbers, also k ..."
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Abstract. We show that there are plenty of infinite sets that satisfy the omniscience principle, in a minimalistic setting for constructive mathematics that is compatible with classical mathematics. A first example of an omniscient set is the onepoint compactification of the natural numbers, also known as the generic convergent sequence. We relate this to Grilliot’s and Ishihara’s Tricks. We generalize this example to many infinite subsets of the Cantor space. These subsets turn out to be ordinals in a constructive sense, with respect to the lexicographic order, satisfying both a wellfoundedness condition with respect to decidable subsets, and transfinite induction restricted to decidable predicates. The use of simple types allows us to reach any ordinal below ɛ0, and richer type systems allow us to get higher. §1. Introduction. We show that there are plenty of infinite sets X that satisfy the omniscience principle for every function p: X → 2, ∃x ∈ X(p(x) = 0) ∨ ∀x ∈ X(p(x) = 1). For X finite this is trivial, and for X = N, this is LPO, the limited principle of omniscience, which of course is and will remain a taboo in any variety of
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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An axiomatic constructive development of the theory of nearness and apartness of a point and a set is introduced as a setting for topology.
Bounded variation implies regulated: A constructive proof. The Journal of symbolic logic
"... Abstract. It is shown constructively that a strongly extensional function of bounded variation on an interval is regulated, in a sequential sense that is classically equivalent to the usual one. This paper continues the constructive study of monotone functions and functions of bounded variation, beg ..."
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Abstract. It is shown constructively that a strongly extensional function of bounded variation on an interval is regulated, in a sequential sense that is classically equivalent to the usual one. This paper continues the constructive study of monotone functions and functions of bounded variation, begun in [3] and [4] (see also [5]). It can be read by anyone who appreciates the distinction between classical and intuitionistic logic, and does not require a detailed knowledge of the constructive theory of R, let alone any abstract constructive analysis. However, the reader will find it helpful to have at hand a copy of [1], [2], [6], or [10]. Throughout the paper, I will be a proper interval in R, Y a metric space,1 and f: I → Y a mapping that is strongly extensional in the sense that ∀x∀y (f(x) 6 = f(y) ⇒ x 6 = y), where, for two elements x, y of a metric space, x 6 = y means ρ(x, y)> 0. It is shown in [4] that an increasing function f: I → R is strongly extensional, and that for all applicable x ∈ I the real numbers f(x−) = limt→x − f(x) and f(x+) = limt→x+ f(t)
Principles Weaker than BDN
, 2012
"... BDN is a weak principle of constructive analysis. Several interesting principles implied by BDN have already been identified, namely the closure of the antiSpecker spaces under product, the Riemann Permutation Theorem, and the Cauchyness of all partially Cauchy sequences. Here these are shown to ..."
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BDN is a weak principle of constructive analysis. Several interesting principles implied by BDN have already been identified, namely the closure of the antiSpecker spaces under product, the Riemann Permutation Theorem, and the Cauchyness of all partially Cauchy sequences. Here these are shown to be strictly weaker than BDN, yet not provable in set theory alone under constructive logic.