Set-theoretic forcing over sites? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T09:37:17Z http://mathoverflow.net/feeds/question/13480 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/13480/set-theoretic-forcing-over-sites Set-theoretic forcing over sites? Mike Shulman 2010-01-30T16:39:56Z 2010-01-30T22:36:45Z <p>All texts I have read on set-theoretic independence proofs consider several different sorts of constructions separately, such as Boolean-valued models (equivalently, forcing over posets), permutation models, and symmetric models. However, the topos-theoretic analogues of these notions&mdash;namely topoi of sheaves on locales, continuous actions of groups, and combinations of the two&mdash;are all special cases of one notion, namely the topos of sheaves on a site. Is there anywhere to be found a direct construction, in the classical world of membership-based set theory, of a "forcing model" relative to an arbitrary site?</p> http://mathoverflow.net/questions/13480/set-theoretic-forcing-over-sites/13488#13488 Answer by François G. Dorais for Set-theoretic forcing over sites? François G. Dorais 2010-01-30T17:44:23Z 2010-01-30T22:36:45Z <p><strike>To the best of my knowledge, this has never been "officially" described in the set theoretic literature.</strike> This has been described by Blass and Scedrov in <em>Freyd's models for the independence of the axiom of choice</em> (Mem. Amer. Math. Soc. 79, 1989). (It is of course implicit and sometimes explicit in the topoi literature, for example Mac Lane and Moerdijk do a fair bit of the translation in <em>Sheaves in Geometry and Logic</em>.) There are certainly a handful of set theorists that are well aware of the generalization and its potential, but I've only seen a few instances of crossover. In my humble opinion, the lack of such crossovers is a serious problem (for both parties). To be fair, there are some important obstructions beyond the obvious linguistic differences. Foremost is the fact that classical set theory is very much a classical theory, which means that the double-negation topology on a site is, to a certain extent, the only one that makes sense for use classical set theory. On the other hand, although very important, the double-negation topology is not often a focal point in topos theory.</p> <p><hr /></p> <p>Thanks to the comments by Joel Hamkins, it appears that there is an even more serious obstruction. In view of the main results of Grigorieff in <em>Intermediate submodels and generic extensions in set theory</em>, Ann. Math. (2) 101 (1975), it looks like the forcing posets are, up to equivalence, precisely the small sites (with the double-negation topology) that preserve the axiom of choice in the generic extension.</p>