Tbilisi Mathematical Journal

Dependent $T$ and existence of limit models

Saharon Shelah

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


Does the class of linear orders have (one of the variants of) the so called $(\lambda,\kappa)$-limit model? It is necessarily unique, and naturally assuming some instances of G.C.H. we get some positive results. More generally, letting $T$ be a complete first order theory and for simplicity assume G.C.H., for regular $\lambda > \kappa > |T|$ does $T$ have (variants of) a $(\lambda,\kappa)$-limit models, except for stable $T$? For some, yes, the theory of dense linear order, for some, no. Moreover, for independent $T$ we get negative results. We deal more with linear orders.

Article information

Tbilisi Math. J., Volume 7, Issue 1 (2014), 99-128.

Received: 29 September 2009
Accepted: 20 November 2014
First available in Project Euclid: 12 June 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03C45: Classification theory, stability and related concepts [See also 03C48]
Secondary: 03C55: Set-theoretic model theory 06A05: Total order

Model theory classification theory dependent theories limit models linear order


Shelah, Saharon. Dependent $T$ and existence of limit models. Tbilisi Math. J. 7 (2014), no. 1, 99--128. doi:10.2478/tmj-2014-0010. https://projecteuclid.org/euclid.tbilisi/1528768956

Export citation


  • Ryszard Engelking and Monika Karłowicz. Some theorems of set theory and their topological consequences. Fundamenta Math., 57:275–285, 1965.
  • Wilfrid Hodges. Model theory, volume 42 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 1993.
  • Bjarni Jónsson. Universal relational systems. Mathematica Scandinavica, 4:193–208, 1956.
  • Bjarni Jónsson. Homogeneous universal relational systems. Mathematica Scandinavica, 8:137–142, 1960.
  • Menachem Kojman and Saharon Shelah. Non-existence of Universal Orders in Many Cardinals. Journal of Symbolic Logic, 57:875–891, 1992. math.LO/9209201.
  • Džamonja Mirna. Club guessing and the universal models. Notre Dame J. Formal Logic, 46:283–300, 2005.
  • M. D. Morley and R. L. Vaught. Homogeneous and universal models. Mathematica Scandinavica, 11:37–57, 1962.
  • Saharon Shelah. Abstract elementary classes near $\aleph_1$. Chapter I. Classification theory of abstract elementary classes. Vol. 18, College Publications, 2009. 0705.4137.
  • Saharon Shelah. Dependent dreams: recounting types. Preprint. 1202.5795.
  • Saharon Shelah. Dependent theories and the generic pair conjecture. Communications in Contemporary Mathematics, in press. math.LO/0702292.
  • Saharon Shelah. Half dependent theories $T$: Recounting in inaccessibles. In preparation.
  • Saharon Shelah. When first order $T$ has limit models. Colloquium Mathematicum. 126:187–204, 2012. math.LO/0603651.
  • Saharon Shelah. Independence results. The Journal of Symbolic Logic, 45:563–573, 1980.
  • Saharon Shelah. Classification of nonelementary classes. II. Abstract elementary classes. In Classification theory (Chicago, IL, 1985), volume 1292 of Lecture Notes in Mathematics, pages 419–497. Springer, Berlin, 1987. Proceedings of the USA–Israel Conference on Classification Theory, Chicago, December 1985; ed. Baldwin, J.T.
  • Saharon Shelah. Classification theory and the number of nonisomorphic models, volume 92 of Studies in Logic and the Foundations of Mathematics. North-Holland Publishing Co., Amsterdam, xxxiv+705 pp, 1990.
  • Saharon Shelah. Classification Theory for Abstract Elementary Classes, volume 18 of Studies in Logic: Mathematical logic and foundations. College Publications, 2009.
  • Saharon Shelah. Dependent first order theories, continued. Israel Journal of Mathematics, 173:1–60, 2009. math.LO/0406440.
  • Saharon Shelah. No limit model in inaccessibles. CRM Proceedings and Lecture Notes, 53:277–290, 2011. 0705.4131.
  • I. Kaplan, N. Lavi and S. Shelah. The generic pair conjecture for dependent finite diagrams. Israel Journal of Mathematics, accepted.