Nonexistence of universal orders in many cardinals

Journal of Symbolic Logic 57 (3):875-891 (1992)
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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 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)

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Citations of this work

Toward classifying unstable theories.Saharon Shelah - 1996 - Annals of Pure and Applied Logic 80 (3):229-255.
On ◁∗-maximality.Mirna Džamonja & Saharon Shelah - 2004 - Annals of Pure and Applied Logic 125 (1-3):119-158.
Universal Structures.Saharon Shelah - 2017 - Notre Dame Journal of Formal Logic 58 (2):159-177.

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References found in this work

Model Theory.Michael Makkai, C. C. Chang & H. J. Keisler - 1991 - Journal of Symbolic Logic 56 (3):1096.
Classification Theory and the Number of Nonisomorphic Models.S. Shelah - 1982 - Journal of Symbolic Logic 47 (3):694-696.
On universal graphs without instances of CH.Saharon Shelah - 1984 - Annals of Pure and Applied Logic 26 (1):75-87.
Simple unstable theories.Saharon Shelah - 1980 - Annals of Mathematical Logic 19 (3):177.

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