NONEXISTENCE OF UNIVERSAL ORDERS IN MANY CARDINALS

Our theme is that not every interesting question in set theory is independent of ZFC. We give an example of a first order theory T with countable D(T) which cannot have a universal model at aleph1 without CH; we prove in ZFC a covering theorem from the hypothesis of the existence of a universal model for some theory; and we prove-again in ZFC-that for a large class of cardinals there is no universal linear order (e.g. in every regular alpeh1 < lambda < 2aleph0). In fact, what we show is that if there is a universal linear order at a regular lambda and its existence is not a result of a trivial cardinal arithmetical reason, then lambda "resembles" aleph1-a cardinal for which the consistency of having a universal order is known. As for singular cardinals, we show that for many singular cardinals, if they are not strong limits then they have no universal linear order. As a result of the nonexistence of a universal linear order, we show the nonexistence of universal models for all theories possessing the strict order property (for example, ordered fields and groups, Boolean algebras, p-adic rings and fields, partial orders, models of PA and so on).