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A note on inequivalence of realizability toposes

Published online by Cambridge University Press:  24 October 2008

P. T. Johnstone  and
E. P. Robinson
P. T. Johnstone
Affiliation:
Department of Pure Mathematics, University of Cambridge, Cambridge CB2 1SB
E. P. Robinson
Affiliation:
Department of Computing and Information Science, Queen's University, Kingston, Ont. K7L 3N6, Canada
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Extract

The general construction of realizability toposes was first described in [3]. The data from which such a topos is constructed (called a partial applicative structure in [3]) consists of a set A equipped with a partial binary operation (called application and denoted by juxtaposition, with the convention that unbracketed expressions are evaluated from left to right) and two constants K, S satisfying

for all x, y, z ∈ A (where, as usual, an equality between possibly undefined terms means ‘one side is defined iff the other is, and then they are equal’). Following Lambek [6], we now call such a structure a partial Schönfinkel algebra, or a global Schönfinkel algebra if application is defined on the whole of A × A. (The name ‘combinatory algebra’ is also in common use.) We write for the realizability topos constructed from A.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 105 , Issue 1 , January 1989 , pp. 1 - 3
Copyright
Copyright © Cambridge Philosophical Society 1989

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References

REFERENCES

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