The Pure Mathematics Department of Cambridge University has a new electronic preprint server (accessible via our home page at http://www.pmms.cam.ac.uk). The first preprint available may be of interest to people on the categories mailing list: it is C. Butz and P.T. Johnstone: Classifying toposes for first-order theories Abstract: By a classifying topos for a first-order theory $\Bbb T$, we mean a topos $\cal E$ such that, for any topos $\cal F$, models of $\Bbb T$ in $\cal F$ correspond exactly to open geometric morphisms ${\cal F} \rightarrow{\cal E}$. 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 classifying topos of such a theory. We also show that every first-order theory has a conservative extension to one which possesses a classifying topos, and we obtain a Heyting-valued completeness theorem for infinitary first-order logic. For those who would prefer to receive a hard copy of this paper, I shall be bringing a supply with me to the Vancouver meeting. Peter Johnstone
Dear Colleagues, the ps-file of the following preprint is available at the homepage http://www.brics.dk/~butz : Topological Completeness for Higher-Order Logic by Steve Awodey (awodey@cmu.edu), Carsten Butz (butz@brics.dk). Abstract: Using recent results in topos theory, two systems of higher-order logic are shown to be complete with respect to sheaf models over topological spaces---so-called ``topological semantics''. The first is classical higher-order logic, with relational quantification of finitely high type; the second system is a predicative fragment thereof with quantification over functions between types, but not over arbitrary relations. The second theorem applies to intuitionistic as well as classical logic. Best regards, Steve Awodey and Carsten Butz
participants (2)
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Carsten Butz -
Dr. P.T. Johnstone