Journal of Symbolic Logic

Nonexistence of Universal Orders in Many Cardinals

Menachem Kojman and Saharon Shelah

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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 $\aleph_1$ 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 regular $\aleph_1 < \lambda < 2^{\aleph_0}$). 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 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).

Article information

J. Symbolic Logic, Volume 57, Issue 3 (1992), 875-891.

First available in Project Euclid: 6 July 2007

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Universal Model linear order covering numbers club guessing strict order property


Kojman, Menachem; Shelah, Saharon. Nonexistence of Universal Orders in Many Cardinals. J. Symbolic Logic 57 (1992), no. 3, 875--891.

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