Nonexistence of universal orders in many cardinals

Journal of Symbolic Logic 57 (3):875-891 (1992)
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 regular $\aleph_1 < \lambda < 2^{\aleph_0}$). 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 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).
Keywords Universal Model   linear order   covering numbers   club guessing   strict order property
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DOI 10.2307/2275437
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Mirna Džamonja & Saharon Shelah (2004). On ◁∗-Maximality. Annals of Pure and Applied Logic 125 (1-3):119-158.
Saharon Shelah (1996). Toward Classifying Unstable Theories. Annals of Pure and Applied Logic 80 (3):229-255.
Menachem Kojman & Saharon Shelah (2001). Fallen Cardinals. Annals of Pure and Applied Logic 109 (1-2):117-129.
Tetsuya Ishiu (2006). The Saturation of Club Guessing Ideals. Annals of Pure and Applied Logic 142 (1):398-424.

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