 
Summary: Workshop on Realizability Preliminary Version
Local Realizability Toposes and a Modal Logic
for Computability
(Extended Abstract)
Steven Awodey 1
Department of Philosophy, Carnegie Mellon University
Lars Birkedal 1
School of Computer Science, Carnegie Mellon University
Dana S. Scott 1
School of Computer Science, Carnegie Mellon University
Abstract
This work is a step toward developing a logic for types and computation that in
cludes both the usual spaces of mathematics and constructions and spaces from
logic and domain theory. Using realizability, we investigate a configuration of three
toposes, which we regard as describing a notion of relative computability. Attention
is focussed on a certain local map of toposes, which we study first 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.
1 Introduction
We report here on the current status of research on the Logic of Types and
