Empty algebras

[Michael Barr <barr@math.mcgill.ca>]

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/ ]