Hostname: page-component-848d4c4894-wg55d Total loading time: 0 Render date: 2024-05-18T05:24:22.077Z Has data issue: false hasContentIssue false

Nonexistence of universal orders in many cardinals

Published online by Cambridge University Press:  12 March 2014

Menachem Kojman
Department of Mathematics, The Hebrew University, Jerusalem, Israel, E-mail:
Saharon Shelah
Department of Mathematics, The Hebrew University, Jerusalem, Israel, E-mail:


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 ). 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).

Research Article
Copyright © Association for Symbolic Logic 1992

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)



[CK]Chang, C. C. and Keisler, H. J., Model theory, North-Holland, Amsterdam, 1973.Google Scholar
[GrSh 174]Grossberg, R. and Shelah, S., On universal locally finite groups, Israel Journal of Mathematics, vol. 44 (1983), pp. 289302.CrossRefGoogle Scholar
[KjSh 447]Kojman, M. and Shelah, S., The universality spectrum of stable unsuperstable theories, Annals of Pure and Applied Logic (to appear).Google Scholar
[Le]Levy, Azriel, Basic set theory, Springer Verlag, Berlin, 1979.CrossRefGoogle Scholar
[M]Mekler, Alan H., Universal structures in power ℵ1, this Journal, vol. 55 (1990), pp. 466477.Google Scholar
[Sh-a]Shelah, Saharon, Classification theory and the number of non-isomorphic models, North-Holland, Amsterdam, 1978.Google Scholar
[Sh-c]Shelah, Saharon, Classification theory and the number of non-isomorphic models, rev. ed., North-Holland, Amsterdam, 1990.Google Scholar
[Sh-e]Shelah, Saharon, Universal classes, in preparation.Google Scholar
[Sh-g]Shelah, Saharon, Cardinal arithmetic, Oxford University Press, Oxford (to appear).CrossRefGoogle Scholar
[Sh 93]Shelah, Saharon, Simple unstable theories, Annals of Mathematical Logic, vol. 19 (1980), pp. 177204.CrossRefGoogle Scholar
[Sh 100]Shelah, Saharon, Independence results, this Journal, vol. 45 (1980), pp. 563573.Google Scholar
[Sh 175]Shelah, Saharon, On universal araphs without instances of CH, Annals of Pure and Applied Logic, vol. 26(1984), pp. 7587.CrossRefGoogle Scholar
[Sh 175a]Shelah, Saharon, Universal graphs without instances of CH, revised, Israel Journal of Mathematics, vol. 70 (1990), pp. 6981.CrossRefGoogle Scholar
[Sh 400]Shelah, Saharon, Cardinal arithmetic, Chapter 9 [tentatively] in [Sh-g].Google Scholar
[Sh 420]Shelah, Saharon, Advances in cardinal arithmetic, Proceedings of the NATO advanced study institute on finite and infinite combinatorics in sets and logic (Banff, 1991 ) (to appear).Google Scholar