1,041 research outputs found
On some peculiar aspects of the constructive theory of point-free spaces
This paper presents several independence results concerning the topos-valid
and the intuitionistic (generalized) predicative theories of locales. In
particular, certain consequences of the consistency of a general form of
Troelstra's uniformity principle with constructive set theory and type theory
are examined
Constructive Mathematics in Theory and Programming Practice
The first part of the paper introduces the varieties of modern constructive mathematics, concentrating on Bishopâs constructive mathematics(BISH). It gives a sketch of both Myhillâs axiomatic system for BISH and a constructive axiomatic development of the real line R. The second part of the paper focuses on the relation between constructive mathematics and programming, with emphasis on Martin-Lofâs theory of types as a formal system for BISH
Heyting-valued interpretations for Constructive Set Theory
We define and investigate Heyting-valued interpretations for Constructive ZermeloâFrankel set theory (CZF). 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
Ordered Locales
We extend the Stone duality between topological spaces and locales to include order: there is an adjunction between the category of preordered topological spaces satisfying the so-called open cone condition, and the newly defined category of ordered locales. The adjunction restricts to an equivalence of categories between spatial ordered locales and sober T 0-ordered spaces with open cones.</p
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