Skip to main content Accessibility help




We introduce the concept of a locally finite abstract elementary class and develop the theory of disjoint $\left( { \le \lambda ,k} \right)$ -amalgamation) for such classes. From this we find a family of complete ${L_{{\omega _1},\omega }}$ sentences ${\phi _r}$ that a) homogeneously characterizes ${\aleph _r}$ (improving results of Hjorth [11] and Laskowski–Shelah [13] and answering a question of [21]), while b) the ${\phi _r}$ provide the first examples of a class of models of a complete sentence in ${L_{{\omega _1},\omega }}$ where the spectrum of cardinals in which amalgamation holds is other that none or all.



Hide All
[1] Baldwin, J. T., Categoricity, University Lecture Notes, vol. 51, American Mathematical Society, Providence, RI, 2009.
[2] Baldwin, J. T. and Boney, W., The Hanf number for amalgamation and joint embedding in AEC’s , Beyond First Order Model Theory, 2015, submitted.
[3] Baldwin, J. T., Friedman, S., Koerwien, M., and Laskowski, C., Three red herrings around Vaught’s conjecture . Transactions of the American Mathematical Society, vol. 368 (2016), pp. 36733694.
[4] Baldwin, J. T., Koerwien, M., and Souldatos, I., The joint embedding property and maximal models , Archive for Mathematical Logic, vol. 55 (2016), no. 3, 545565.
[5] Baldwin, J. T. and Kolesnikov, A., Categoricity, amalgamation, and tameness . Israel Journal of Mathematics, vol. 170 (2009), pp. 411443.
[6] Baldwin, J. T., Kolesnikov, A., and Shelah, S., The amalgamation spectrum, this JOURNAL, vol. 74 (2009), pp. 914928.
[7] Baldwin, J. T., Laskowski, M. C., and Shelah, S., Constructing many uncountable atomic models in ${\aleph _1}$ , this JOURNAL, vol. 81 (2016), pp. 11421162.
[8] Baldwin, J. T. and Souldatos, I., Complete ${L_{{\omega _1},\omega }}$ -sentences with maximal models in multiple cardinalities, 2015, submitted.
[9] Button, T. and Walsh, S., Ideas and results in model theory: Reference, realism, structure and categoricity, 2015, manuscript,
[10] Hart, B. and Shelah, S., Categoricity over P for first order T or categoricity for $\phi \in {{\rm{l}}_{{\omega _1}\omega }}$ can stop at ${\aleph _k}$ while holding for ${\aleph _0}, \ldots ,{\aleph _{k - 1}}$ . Israel Journal of Mathematics, vol. 70 (1990), pp. 219235.
[11] Hjorth, G., Knight’s model, its automorphism group, and characterizing the uncountable cardinals . Journal of Mathematical Logic, vol. 2 (2002), pp. 113144.
[12] Kolesnikov, A. and Lambie-Hanson, C., Hanf numbers for amalgamation of coloring classes, 2014, preprint.
[13] Laskowski, M. C. and Shelah, S., On the existence of atomic models, this JOURNAL, vol. 58 (1993), pp. 11891194.
[14] Malitz, J., The Hanf number for complete ${L_{{\omega _1},\omega }}$ sentences, The Syntax and Semantics of Infinitary Languages (Barwise, J., editor), LNM 72, Springer-Verlag, Heidelberg, 1968, pp. 166181.
[15] Shelah, S., Classification theory for nonelementary classes. I. the number of uncountable models of $\psi \in {L_{{\omega _1}\omega }}$ part A. Israel Journal of Mathematics, vol. 46 (1983), no. 3, pp. 212240.
[16] Shelah, S., Classification theory for nonelementary classes. II. the number of uncountable models of $\psi \in {L_{{\omega _1}\omega }}$ part B. Israel Journal of Mathematics, vol. 46 (1983), no. 3, pp. 241271.
[17] Shelah, S., Classification Theory for Abstract Elementary Classes, Studies in Logic, College Publications, London, 2009.
[18] Shelah, S., Classification Theory for Abstract Elementary Classes: II. Studies in Logic, College Publications, London, 2010.
[19] Shelah, S., Classification of nonelementary classes II, abstract elementary classes , Classification Theory (Chicago, IL, 1985), Proceedings of the USA–Israel Conference on Classification Theory (Baldwin, J. T., editor), Lecture Notes in Mathematics, vol. 1292, Springer, Berlin, 1987, pp. 419497.
[20] Souldatos, I., Characterizing the powerset by a complete (Scott) sentence . Fundamenta Mathematica, vol. 222 (2013), pp. 131154.
[21] Souldatos, I., Notes on cardinals that are characterizable by a complete (Scott) sentence . Notre Dame Journal of Formal Logic, vol. 55 (2013), pp. 533551.
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: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed