Sh:409

• Kojman, M., & Shelah, S. (1992). Nonexistence of universal orders in many cardinals. J. Symbolic Logic, 57(3), 875–891. arXiv: math/9209201 DOI: 10.2307/2275437 MR: 1187454
• Abstract:
  We give an example of a first order theory T with countable D(T) which cannot have a universal model at \aleph_1 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 \aleph_1< \lambda< 2^{\aleph_0}). In fact, what we show is that if there is a universal linear order at a regular \lambda and its existence is not a result of a trivial cardinal arithmetical reason, then \lambda “resembles” \aleph_1 – 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 non existence of a universal linear order, we show the non-existence 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).
• Version 1993-08-29_10 (19p) published version (18p)

@article{Sh:409,
 author = {Kojman, Menachem and Shelah, Saharon},
 title = {{Nonexistence of universal orders in many cardinals}},
 journal = {J. Symbolic Logic},
 fjournal = {The Journal of Symbolic Logic},
 volume = {57},
 number = {3},
 year = {1992},
 pages = {875--891},
 issn = {0022-4812},
 mrnumber = {1187454},
 mrclass = {03C55 (03C45 03C50 06A05)},
 doi = {10.2307/2275437},
 note = {\href{https://arxiv.org/abs/math/9209201}{arXiv: math/9209201}},
 arxiv_number = {math/9209201}
}