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0 9 r e v i s i o n : 1 9 9 3 0 8 2 9 m o d i f i e d : 1 9 9 3 0 8 2 9 Non-existence of Universal Orders in Many Cardinals Menachem Kojman and Saharon Shelah† Department of Mathematics, Hebrew University 91904 Jerusalem, Israel ABSTRACT 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 א1 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 א1 < λ < 2 0). In fact, what we show is that if there is a universal linear order at a regular λ and its existence is not a result of a trivial cardinal arithmetical reason, then λ “resembles” א1 — 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 non existence of a universal linear order, we show the non-existence 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).