Non-existence of Universal Orders in Many Cardinals

by Kojman and Shelah. [KjSh:409]
J Symbolic Logic, 1992
We give an example of a first order theory T with countable D(T) which cannot have a universal model at aleph_1 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 aleph_1< lambda < 2^{aleph_0}). 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'' aleph_1 -- 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).

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