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DISJOINT AMALGAMATION IN LOCALLY FINITE AEC

  • JOHN T. BALDWIN (a1), MARTIN KOERWIEN (a2) and MICHAEL C. LASKOWSKI (a3)

Abstract

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.

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[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, http://faculty.sites.uci.edu/seanwalsh/files/2015/01/button-walsh-arXiv-submit.1150992.pdf.
[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.
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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
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