Relating first-order set theories, toposes and categories of classes
نویسندگان
چکیده
منابع مشابه
Relating first-order set theories, toposes and categories of classes
Article history: Received 22 December 2010 Received in revised form 12 December 2012 Accepted 12 May 2013 Available online 8 July 2013 MSC: 03E70 03F50 03G30 18B05
متن کاملfirst - order set theories , toposes and categories of classes
This paper introduces Basic Intuitionistic Set Theory BIST, and investigates it as a first-order set theory extending the internal logic of elementary toposes. Given an elementary topos, together with the extra structure of a directed structural system of inclusions (dssi) on the topos, a forcing-style interpretation of the language of first-order set theory in the topos is given, which conserv...
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We show how to interpret the language of first-order set theory in an elementary topos endowed with, as extra structure, a directed structural system of inclusions (dssi). As our main result, we obtain a complete axiomatization of the intuitionistic set theory validated by all such interpretations. Since every elementary topos is equivalent to one carrying a dssi, we thus obtain a first-order s...
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By a classifying topos for a first-order theory T, we mean a topos E such that, for any topos F , models of T in F correspond exactly to open geometric morphisms F → E . We show that not every (infinitary) first-order theory has a classifying topos in this sense, but we characterize those which do by an appropriate ‘smallness condition’, and we show that every Grothendieck topos arises as the c...
متن کاملOn Relating Type Theories and Set Theories
Introduction The original motivation 1 for the work described in this paper was to determine the proof theoretic strength of the type theories implemented in the proof development systems Lego and Coq, Luo and Pollack 92, Barras et al 96]. These type theories combine the impredicative type of propositions 2 , from the calculus of constructions, Coquand 90], with the inductive types and hierarch...
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ژورنال
عنوان ژورنال: Annals of Pure and Applied Logic
سال: 2014
ISSN: 0168-0072
DOI: 10.1016/j.apal.2013.06.004