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Constructive Order Completeness
, 2004
"... Partially ordered sets are investigated from the point of view of Bishop’s constructive mathematics. Unlike the classical case, one cannot prove constructively that every nonempty bounded above set of real numbers has a supremum. However, the order completeness of R is expressed constructively by an ..."
Abstract

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Partially ordered sets are investigated from the point of view of Bishop’s constructive mathematics. Unlike the classical case, one cannot prove constructively that every nonempty bounded above set of real numbers has a supremum. However, the order completeness of R is expressed constructively by an equivalent condition for the existence of the supremum, a condition of (upper) order locatedness which is vacuously true in the classical case. A generalization of this condition will provide a definition of upper locatedness for a partially ordered set. It turns out that the supremum of a set S exists if and only if S is upper located and has a weak supremum—that is, the classical least upper bound. A partially ordered set will be called order complete if each nonempty subset that is bounded above and upper located has a supremum. It can be proved that, as in the classical mathematics, R n is order complete. 1
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.
Constructive Suprema 1
"... Abstract: Partially ordered sets are investigated from the point of view of Bishop’s constructive mathematics, which can be viewed as the constructive core of mathematics and whose theorems can be translated into many formal systems of computable mathematics. The relationship between two classically ..."
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Abstract: Partially ordered sets are investigated from the point of view of Bishop’s constructive mathematics, which can be viewed as the constructive core of mathematics and whose theorems can be translated into many formal systems of computable mathematics. The relationship between two classically equivalent notions of supremum is examined in detail. Whereas the classical least upper bound is based on the negative concept of partial order, the other supremum is based on the positive notion of excess relation. Equivalent conditions of existence are obtained for both suprema in the general case of a partially ordered set; other equivalent conditions are obtained for subsets of a lattice and, in particular, for subsets of R n.