Topos theory and consistent histories: The internal logic of the set of all consistent sets
 C. J. Isham
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Abstract
A major problem in the consistenthistories approach to quantum theory is contending with the potentially large number of consistent sets of history propositions. One possibility is to find a scheme in which a unique set is selected in some way. However, in this paper the alternative approach is considered in which all consistent sets are kept, leading to a type of ‘manyworldviews’ picture of the quantum theory. It is shown that a natural way of handling this situation is to employ the theory of varying sets (presheafs) on the spaceB of all nontrivial Boolean subalgebras of the orthoalgebraUP of history propositions. This approach automatically includes the feature whereby probabilistic predictions are meaningful only in the context of a consistent set of history propositions. More strikingly, it leads to a picture in which the ‘truth values’ or ‘semantic values’ of such contextual predictions are not just twovalued (i.e., true and false) but instead lie in a larger logical algebra—a Heyting algebra—whose structure is determined by the spaceB of Boolean subalgebras ofUP. This topostheoretic structure thereby gives a coherent mathematical framework in which to understand the internal logic of the manyworldviews picture that arises naturally in the approach to quantum theory based on the ideas of consistent histories.
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 Title
 Topos theory and consistent histories: The internal logic of the set of all consistent sets
 Journal

International Journal of Theoretical Physics
Volume 36, Issue 4 , pp 785814
 Cover Date
 19970401
 DOI
 10.1007/BF02435786
 Print ISSN
 00207748
 Online ISSN
 15729575
 Publisher
 Kluwer Academic PublishersPlenum Publishers
 Additional Links
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 Authors

 C. J. Isham ^{(1)}
 Author Affiliations

 1. Blackett Laboratory, Imperial College of Science, Technology and Medicine, SW7 2BZ, London, UK