<<Could someone please tell me where the term "Grothendieck universes" comes from? Some books connect Grothendieck's name with this particular approach to avoid set-theoretical difficulties, others don't. What is the current status on these foundational matters anyway? Has anything else emerged besides MacLanes one universe approach, or Feferman's use of reflection principles?>> I don't have the Grothendieck reference, but it is not attributed to him by accident. Hopefully someone will supply particulars. I enter this to observe (1) One other thing that has emerged is Blass' position, that it doesn't matter (assuming we wish to do mathematics, not theory of languages). That is clearly and effectively stated by Barr and Wells in the preface or whatever it is to <Toposes Triples and Theories>. Note further that this treatment of the <foundational difficulties> is in the spirit of Grothendieck. MacLane, sorry, Mac Lane is in the spirit of Plato: the one real universe is out there and we should be discovering its properties. It has more impressive adherents than Plato, e.g. G"odel, but it is not viable. John Isbell