• Corpus ID: 119177789

Unified Approach to Real Numbers in Various Mathematical Settings

  title={Unified Approach to Real Numbers in Various Mathematical Settings},
  author={Davorin Levsnik},
We provide a setting-independent definition of reals by introducing the notion of a streak. We show that various standard constructions of reals satisfy our definition. We study the structure of reals by noting that its pieces correspond to reflections on the category of streaks. 



Real numbers and other completions

A notion of completeness and completion suitable for use in the absence of countable choice is developed. This encompasses the construction of the real numbers as well as the completion of an

A universal characterization of the closed Euclidean interval

  • M. EscardóA. Simpson
  • Mathematics
    Proceedings 16th Annual IEEE Symposium on Logic in Computer Science
  • 2001
A notion of interval object in a category with finite products is proposed, providing a universal property for closed and bounded real line segments, and it is proved that an interval object exists in and elementary topos with natural numbers object.

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The aim of this book is to construct categories of spaces which contain all the C?-manifolds, but in addition infinitesimal spaces and arbitrary function spaces. To this end, the techniques of

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The thesis presents the subject of synthetic topology, especially with relation to metric spaces. A model of synthetic topology is a categorical model in which objects possess an intrinsic topology

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It is consistent with constructive set theory (without Countable Choice, clearly) that the Cauchy reals (equivalence classes of Cauchy sequences of rationals) are not Cauchy complete. Related results

The Dedekind reals in abstract Stone duality

The core of the paper constructs the real line using two-sided Dedekind cuts, and shows that the closed interval is compact and overt, where these concepts are defined using quantifiers.

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Preface Acknowledgements Foreword Introduction 1. A primer on ordered sets and lattices 2. Order theory of domains 3. The Scott topology 4. The Lawson Topology 5. Morphisms and functors 6. Spectral

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Introduction 1. Basic features of smooth worlds 2. Basic differential calculus 3. First applications of the differential calculus 4. Applications to physics 5. Multivariable calculus and applications

Constructive set theory

  • J. Myhill
  • Philosophy
    Journal of Symbolic Logic
  • 1975
There is a widespread current impression that the theory of Godel functionals, with quantifiers and choice, is the appropriate formalism for Bishop's book [1], but this is not so, and in more advanced mathematics the complexities become intolerable.