Dear Michael, What Vaughan called the "clean-shaven gods" argument is, I think, the normal logicians' excuse for barring empty carriers. The logical deduction (all) x. P(x) ------------- P(a) ------------- (exists) x. P(x) is false for empty carriers, but many logicians persuade themselves that the deduction is correct and the empty carriers are therefore monsters. Vaughan I think is taking P(x) as "x is clean-shaven" in the theory of gods. You can get a similar phenomenon with equational reasoning, especially for many-sorted algebras. I expect this is what bothered Meinke and Tucker when they wrote their monograph on universal algebra. The fix was discovered by Mostowski (I believe). To use a free variable such as a is to hypothesize a denotation for it in the carrier. Therefore in each deduction one should be explicit about the context of free variables that may be used. This deals cleanly with empty carriers. All the best, Steve. Michael Barr wrote:

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

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