Dear Bob: Here is a distillation of the answers I got from several people on my question on set theory. What I was asking for (eventually) is called a weakly inaccessible cardinal. What Dan Leivant constructed was actually not regular either (Les Nelson). If it was it was inaccessible. There are models of set theory with no cardinals that satisfy the GCH and probably (Blass wasn't certain) had no weakly inaccessibles either. Oddly enough those models were constructed using some large cardinals. No stranger, actually, than Freyd's construction of models lacking choice that start from ZFC and build an elementary topos lacking choice and then a model of set theory also lacking choice. So the upshot is that what I wanted is indeed weaker than the existence of inaccessibles, but still fairly strong. Fortunately, it is only a peripheral to what I was doing. It clarified a side issue that had no effect on the main result, which is that you certainly don't have to delve into the mysteries of non-well-founded sets in order to show that the coalgebras for an inaccessibly accessible functor have a terminal model no larger than the inaccessible provided the functor takes smaller cardinals to smaller cardinals. Michael