Skip to main content
    • Aa
    • Aa

Nonexistence of universal orders in many cardinals

  • Menachem Kojman (a1) and Saharon Shelah (a2)

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).

Linked references
Hide All

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.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Full text views

Total number of HTML views: 0
Total number of PDF views: 2 *
Loading metrics...

Abstract views

Total abstract views: 37 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 24th July 2017. This data will be updated every 24 hours.