(I promised Bob I'd give it a rest (my idea, not his), but I can't resist one more shot. From: oliver@math.ucla.edu i) If my model says there are inaccessibles,... my model is clearly better, because it has everything in it that yours does, and more besides; moreover, we haven't lost any nice features that your model might happen to have, Consistency is not a nice feature? News to me. My model is more likely to be consistent than yours. Inconsistency of yours would be a beautiful result to most people, like a beautiful building or park. Inconsistency of mine would be a magnitude-9 earthquake! Furthermore mathematics has had no trouble imagining my model since 1939 when Goedel showed us how. It is *very* hard to imagine your model, so hard that not even the smartest people in the world have been able to do it in over half a century of trying. This is *not* a good sign. Strengthen this intuition to "things we can construct in L" and you start to see why 0# should exist. This is an argument for assuming the existence of everything whose nonexistence you can't actually *prove*. But that forces you to retreat every time we disprove the existence of yet another ordinal. Such discoveries happen periodically, and do not astonish working mathematicians. A much more stable approach would be to accept the easily imagined, and reject what is hard to imagine. Following that strategy, you only have to retreat in the face of 50-year or even 200-year earthquakes. Anyway, what are we arguing about here? What exactly *is* the advantage of assuming inaccessibles? (Let's not tempt fate and get too close to inconsistency by assuming measurables!) Sure you can shorten some contrived proofs a lot with inaccessibles, but can you shorten a proof some mathematician might care about? Or do anything else useful with them? Vaughan Pratt