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Nonexistence of universal orders in many cardinals

Published online by Cambridge University Press:  12 March 2014

Menachem Kojman  and
Saharon Shelah
Menachem Kojman
Affiliation:
Department of Mathematics, The Hebrew University, Jerusalem, Israel, E-mail: kojman@math.huji.ac.il
Saharon Shelah
Affiliation:
Department of Mathematics, The Hebrew University, Jerusalem, Israel, E-mail: shelah@math.huji.ac.il
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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 ). 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).

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 57 , Issue 3 , September 1992 , pp. 875 - 891
Copyright
Copyright © Association for Symbolic Logic 1992

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References

REFERENCES

[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. 289–302.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. 466–477.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. 177–204.CrossRefGoogle Scholar
[Sh 100]Shelah, Saharon, Independence results, this Journal, vol. 45 (1980), pp. 563–573.Google Scholar
[Sh 175]Shelah, Saharon, On universal araphs without instances of CH, Annals of Pure and Applied Logic, vol. 26(1984), pp. 75–87.CrossRefGoogle Scholar
[Sh 175a]Shelah, Saharon, Universal graphs without instances of CH, revised, Israel Journal of Mathematics, vol. 70 (1990), pp. 69–81.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