Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-13T07:19:36.741Z Has data issue: false hasContentIssue false
  • English
  • Français

Classification and interpretation

Published online by Cambridge University Press:  12 March 2014

Andreas Baudisch
Andreas Baudisch*
Affiliation:
Institut für Mathematik, Akademie der Wissenschaften der DDR, 1086 Berlin, East Germany
Get access Rights & Permissions [Opens in a new window]

Abstract

Let S and T be countable complete theories. We assume that T is superstable without the dimensional order property, and S is interpretable in T in such a way that every model of S is coded in a model of T. We show that S does not have the dimensional order property, and we discuss the question of whether Depth(S) ≤ Depth(T). For Mekler's uniform interpretation of arbitrary theories S of finite similarity type into suitable theories TS of groups we show that Depth(S) ≤ Depth(TS) ≤ 1 + Depth(S).

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 54 , Issue 1 , March 1989 , pp. 138 - 159
Copyright
Copyright © Association for Symbolic Logic 1989

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Baudisch, A., On Lascar rank in nonmultidimensional ω-stable theories, Logic Colloquium '85, North-Holland, Amsterdam, 1987, pp. 33–51.CrossRefGoogle Scholar
[2]Berline, C. and Lascar, D., Superstable groups, Annals of Pure and Applied Logic, vol. 30 (1986), pp. 1–43.CrossRefGoogle Scholar
[3]Bouscaren, E., Dimensional order property and pairs of models, Thèse de Doctorat d'État, Université Paris-VII, Paris, 1985 (also Annals of Pure and Applied Logic (to appear)).Google Scholar
[4]Harrington, L. and Makkai, M., An exposition of Shelah's “main gap”: Counting uncountable models of ω-stable and superstable theories, Notre Dame Journal of Formal Logic, vol. 26 (1985), pp. 139–177.CrossRefGoogle Scholar
[5]Hodges, W., Model theory, Chapter VI (in preparation).Google Scholar
[6]Lascar, D., Définissabilité de types en théorie des modèles, Thèse, Université Paris-VII, Paris, 1975.Google Scholar
[7]Lascar, D., Ordre de Rudin-Keisler et poids dans les théories ω-stables, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 28 (1982), pp. 411–430.CrossRefGoogle Scholar
[8]Lascar, D., Quelques précisions sur la d. o. p. et la profondeur d'une théorie, this Journal, vol. 50(1985), pp. 316–330.Google Scholar
[9]Lascar, D., Stabilité en théorie des modèles, Institut de Mathématique Pure et Appliquée, Université Catholique de Louvain, and Cabay Libraire-Éditeur S. A., Louvain-La-Neuve, 1986.Google Scholar
[10]Lascar, D., Les groupes ω-stables de rang fini, Transactions of the American Mathematical Society, vol. 292 (1985), pp. 451–462.Google Scholar
[11]Lascar, D. and Poizat, B., An introduction to forking, this Journal, vol. 44(1979), pp. 330–350.Google Scholar
[12]Makkai, M., A survey of basic stability theory, with particular emphasis on orthogonality and regular types, Israel Journal of Mathematics, vol. 49 (1984), pp. 181–238.CrossRefGoogle Scholar
[13]Mekler, A. H., Stability of nilpotent groups of class 2 and prime exponent, this Journal, vol. 46 (1981), pp. 781–788.Google Scholar
[14]Poizat, B., Cours de théorie de modèles, Nur al-Mantiq wal-Ma'rifah, Villeurbanne, 1985.Google Scholar
[15]Poizat, B., À propos de groupes stables, Logic Colloquium '85, North-Holland, Amsterdam, 1987, pp. 245–265.CrossRefGoogle Scholar
[16]Saffe, J., The number of uncountable models of ω-stable theories, Annals of Pure and Applied Logic, vol. 24 (1983), pp. 231–261.CrossRefGoogle Scholar
[17]Shelah, S., Classification theory and the number of non-isomorphic models, North-Holland, Amsterdam, 1978.Google Scholar
[18]Shelah, S., The spectrum problem. I: ℵε-saturated models, the main gap, Israel Journal of Mathematics, vol. 43 (1982), pp. 324–356.CrossRefGoogle Scholar
[19]Shelah, S., The spectrum problem. II: Totally transcendental theories, Israel Journal of Mathematics, vol. 43 (1982), pp. 357–364.CrossRefGoogle Scholar
[20]Shelah, S., Classification of first order theories which have a structure theorem, Bulletin (New Series) of the American Mathematical Society, vol. 12 (1985), pp. 227–232.CrossRefGoogle Scholar
[21]Shelah, S., Harrington, L., and Makkai, M., A proof of Vaught's conjecture for totally transcendental theories, Israel Journal of Mathematics, vol. 49 (1984), pp. 259–280.CrossRefGoogle Scholar
[22]Delon, F., Types localement isolés et théorème des deux cardinaux dans les théories stables dénombrables, Groupe d'Étude de Théories Stables, Ire Année: 1977/78, Secrétariat Mathématique, Paris, 1978, Exposé 7.Google Scholar