I have just had a chance to read what Saunders wrote. I think there are a number of good observations there. I just wanted to add something on the question of empty models. I have tried to get UAists to explain the reason to me and there was only reason that had any force and that was that an ultraproduct ``should'' be empty iff the set of coordinates in which it is empty is large. But the usual definition as a quotient of a product does not have that property since it is empty as soon as one coordinate is. But there is another definition that is equivalent, UNLESS some coordinate is empty and that is as a (filtered) colimit of products, taken over large sets of indices. This works in all cases and has the desired properties when some factors are empty. --Michael ++++++++++++++++++++++++++++++++++++++++