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Nonexistence of universal orders in many cardinals

  • Menachem Kojman (a1) and Saharon Shelah (a2)
  • DOI: http://dx.doi.org/10.2307/2275437
  • Published online: 01 March 2014
Abstract
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).

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This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[GrSh 174]R. Grossberg and S. Shelah , On universal locally finite groups, Israel Journal of Mathematics, vol. 44 (1983), pp. 289302.

[Le]Azriel Levy , Basic set theory, Springer Verlag, Berlin, 1979.

[Sh 93]Saharon Shelah , Simple unstable theories, Annals of Mathematical Logic, vol. 19 (1980), pp. 177204.

[Sh 175]Saharon Shelah , On universal araphs without instances of CH, Annals of Pure and Applied Logic, vol. 26(1984), pp. 7587.

[Sh 175a]Saharon Shelah , Universal graphs without instances of CH, revised, Israel Journal of Mathematics, vol. 70 (1990), pp. 6981.

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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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