I just wanted to comment on the "empty algebra" business. Every person conversant in CT ought to know this. I had never heard the "clean-shaven gods" argument, but another substantive reason logicians give for avoiding empty models is that it messes up ultraproducts. Specifically, an ultraproduct will be empty if even one factor is. This can be totally avoided by using a better definition of ultraproduct. Simply define as the (directed) colimit, taken over all the sets in the ultrafilter, of all the large products. Now the ultraproduct will be empty iff the set of empty factors is large. Incidentally, I like the idea of the -1 cell. It means that the Euler characteristic of any simplex (and any contractible space) is 0, which is the number of holes. But it also makes the whole detour into "reduced homology" unnecessary. Michael [For admin and other information see: http://www.mta.ca/~cat-dist/ ]