The Dedekind Reals in Abstract Stone Duality
- 26 July 2005
With Paul Taylor.
Abstract: Abstract Stone Duality (ASD) is an approach to topology that provides an abstract and conceptually satisfying account of topological spaces. The calculus of ASD reveals the computational content of various topological notions and suggests how to compute with them. The distinguishing feature of ASD is a direct axiomatisation in terms of spaces and maps, which does not rely on an underlying set-theoretic or topos-theoretic foundation.
This paper makes the first step in real analysis within ASD, namely the construction of the real line using two-sided Dedekind cuts. Compactness and overtness of the closed interval are proved, and the arithmetic operations are defined. The ASD calculus gives programs for computing the arithmetic operations and the quantifiers that express compactness and overtness.
As the paper aims to be a self-contained introduction to ASD for those interested in constructive and computable topology and analysis, it includes a rapid survey of the ASD calculus. The foundational background to the calculus was covered in detail in earlier work.
Further topics in real analysis within ASD, such as the Intermediate Value Theorem, are presented in a separate paper by Paul Taylor which builds on this one.
To be presented at Computability and Complexity in Analysis 2005, Kyoto, Japan.
Download: an up-to-date version from Paul Taylor's Abstract Stone Duality page.
Dear Andrej, Comes very close to my intuition of spacetime. This is considered to be a dual manifestatin of a finite metaphysical logical space defined by a simple mathematical object known sinc many decades. A perfect correlation between the internal logic, geomatrically visualized by this mathematical object, and the realization of Nature on the fundamental scale is found. The Everything is found to be the dual physical manifestation of an a priori existing logical space - like thought and its realization. A logical space does not have a fabric, is somehow a continuum that could get a logical structure when expressing some context.
That is why also spacetime, being its duality, should not have a fabric/ metric. In connection with a new orbifold-substructured dynamic superstring concept it was found that these dynamic superstring entities holographically express themselves in the surrounding spacetime continuum. This implies that spacetime gets a fabric / structure locally by the events that take place in it. The spacetime continuum does not a priori have a fabric, but gets it by events. The kind of fabric is geometrically defined according to what is expressed - the particle, gravity, fields ....
That is why I am so much interested. But my is pure intuition. I would be pleased to get a reply
@Paul: you just use a lot of buzzwords, and what you wrote could be written as a comment to any blog post about spacetime. So sorry, I don't have a real reply.
The links to Paul's site are broken, he has a new domain.