Local Realizability Toposes and a Modal Logic for Computability

Abstract

This work is a step toward the development of a logic for types and computation that includes not only the usual spaces of mathematics and constructions, but also spaces from logic and domain theory. Using realizability, we investigate a configuration of three toposes that we regard as describing a notion of relative computability. Attention is focussed on a certain local map of toposes, which we first study axiomatically, and then by deriving a modal calculus as its internal logic. The resulting framework is intended as a setting for the logical and categorical study of relative computability

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Author Profiles

Dana Scott
Carnegie Mellon University
Steve Awodey
Carnegie Mellon University

Citations of this work

Epistemic Modality and Hyperintensionality in Mathematics.Timothy Bowen - 2017 - Dissertation, Arché, University of St Andrews
Hyperintensional Ω-Logic.Hasen Khudairi - 2019 - In Matteo Vincenzo D'Alfonso & Don Berkich (eds.), On the Cognitive, Ethical, and Scientific Dimensions of Artificial Intelligence. Springer Verlag. pp. 65-82.
Sheaf toposes for realizability.Steven Awodey & Andrej Bauer - 2008 - Archive for Mathematical Logic 47 (5):465-478.
Introduction to Turing categories.J. Robin B. Cockett & Pieter Jw Hofstra - 2008 - Annals of Pure and Applied Logic 156 (2):183-209.

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