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Class forcing and topos theory
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Title  Class forcing and topos theory 
Title of Series  Topos à l'IHES 
Part Number  12 
Number of Parts  28 
Author 
Roberts, David

License 
CC Attribution 3.0 Unported: You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor. 
DOI  10.5446/20743 
Publisher  Institut des Hautes Études Scientifiques (IHÉS) 
Release Date  2015 
Language  English 
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Subject Area  Mathematics 
Abstract  It is wellknown that forcing over a model of material set theory co rresponds to taking sheaves over a small site (a poset, a complete Boolean algebra, and so on). One phenomenon that occurs is that given a small site, all new subsets created are smaller than a fixed bound depending on the size of the site. There is a more general notion of forcing invented by Easton to create new subsets of arbitrarily large sets, namely class forcing, where one starts with a partially ordered class. The existing theory of class forcing is entirely classical, with no corresponding intuitionist theory as in ordinary forcing. Our understanding of its relation to topos theory is in its infancy, but it is clear that class forcing is about taking small sheaves on a large site. That these do not automatically form a topos means that the theory has interesting twists and turns. This talk will outline the theory of class forcing from a category/topos point of view, give examples and constructions, and fin ally a list of open questions – not least being whether an intuitionistic version of Easton’s theorem on the continuum function holds. 
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