Hostname: page-component-7f64f4797f-d7bbv Total loading time: 0 Render date: 2025-11-03T11:13:57.755Z Has data issue: false hasContentIssue false
  • English
  • Français

Upward categoricity from a successor cardinal for tame abstract classes with amalgamation

Published online by Cambridge University Press:  12 March 2014

Olivier Lessmann
Olivier Lessmann*
Affiliation:
Mathematical Institute, Oxford University, Oxford, OX1 3LB., U.K., E-mail: lessniann@maths.ox.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper is devoted to the proof of the following upward categoricity theorem: Let be a tame abstract elementary class with amalgamation, arbitrarily large models, and countable Löwenheim-Skolem number. If is categorical in ℵ then is categorical in every uncountable cardinal. More generally, wc prove that if is categorical in a successor cardinal λ+ then is categorical everywhere above λ+.

Information

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 70 , Issue 2 , June 2005 , pp. 639 - 660
Copyright
Copyright © Association for Symbolic Logic 2005

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.)

Article purchase

Temporarily unavailable

References

REFERENCES

[Ba] Baldwin, John T., Categoricity, Online book on nonelementary classes. Available at http://www2.math.uic.edu/~jbaldwin/pub/AEClec.pdf.Google Scholar
[Ba1] Baldwin, John T., Collected preprints at http://www2.math.uic.edu/~jbaldwin/model.html.Google Scholar
[BaLe] Baldwin, John T. and Lessmann, Olivier, Amalgamation properties and finite models in Ln-theories, Archive for Mathematical Logic, vol. 41 (2002), no. 2, pp. 155–167.Google Scholar
[BaSh] Baldwin, John T. and Shelah, Saharon, Examples of nonlocality, in prepatarion.Google Scholar
[Dj] DjordjeviĆ, Marko, Finite variable logic, stability and finite models, this Journal, vol. 66 (2001), no. 2, pp. 837–858.Google Scholar
[Ga] Gavrilovich, Misha, Covers of algebraic varieties, Preprint.Google Scholar
[Gr] Grossberg, Rami, Classification theory for abstract elementary classes, Logic and algebra (Zhang, Yi, editor), Contemporary Mathematics, vol. 302, American Mathematical Society, Providence, RI, 2002, pp. 165–204.Google Scholar
[GrKo] Grossberg, Rami and Kolesnikov, Alexei, Excellent classes are tame, Preprint, available at http://www.math.cmu.edu/-rarni/.Google Scholar
[GrVD] Grossberg, Rami and Vandieren, Monica, Upward categoricity transfer theorem for tame abstract elementary classes, Submitted, available at http://www.math.lsa.umich.edu/~mvd/home.html.Google Scholar
[Je] Jech, Thomas, Set theory, Academic Press, New York, 1978.Google Scholar
[Ke] Keisler, H. Jerome, Model theory for infinitary logic, North-Holland Publishing Co., Amsterdam, 1971.Google Scholar
[Le1] Lessmann, Olivier, Dimension theory inside a homogeneous model, Logic colloquium '02 (Münster), 2002, to appear.Google Scholar
[Le2] Lessmann, Olivier, An introduction to excellent classes, Logic and algebra (Zhang, Yi, editor), Contemporary Mathematics, vol. 302, American Mathematical Society, Providence, RI, 2002.Google Scholar
[MeSh] Mekler, Alan H. and Shelah, Saharon, Almost free algebras, Israel Journal of Mathematics, vol. 89 (1995), pp. 237–259.Google Scholar
[Sh3] Shelah, Saharon, Finite diagrams stable in power, Annals of Mathematical Logic, vol. 2 (1970/1971), pp. 69–118.Google Scholar
[Sh87a] Shelah, Saharon, Classification theory for nonelementary classes, I, The number of uncountable models of Ψ ϵ L ω1,ω, Part A, Israel Journal of Mathematics, vol. 46 (1983), pp. 212–240.Google Scholar
[Sh87b] Shelah, Saharon, Classification theory for nonelementary classes, I, The number of uncountable models of Ψ ϵ L ω1,ω, Part B, Israel Journal of Mathematics, vol. 46 (1983), pp. 241–273.Google Scholar
[Sh88] Shelah, Saharon, Classification of nonelementary classes, II, Abstract elementary classes, Classification Theory of Lecture Notes in Mathematics, Lecture Notes in Math., vol. 1292, Springer, Berlin, 1987. pp. 419–497, Categoricity in Abstract Elementary Classes with No Maximal Models.Google Scholar
[Sh394] Shelah, Saharon, Categoricity of abstract classes with amalgamation.Google Scholar
[Sh576] Shelah, Saharon, Categoricity of an abstract elementary class in two successive cardinals, Israel Journal of Mathematics, vol. 126 (2001), pp. 29–128.Google Scholar
[Sh600] Shelah, Saharon, Categoricity in abstract elementary classes: going up an inductive step, Preprint. available at http://shelah.logic.at/.Google Scholar
[Sh702] Shelah, Saharon, On what Ida not understand (and have something to say), model theory, Mathematica Japonica, vol. 51 (2000), pp. 329–377.Google Scholar
[ShVi] Shelah, Saharon and Villaveces, Andrés, Toward categoricity for classes with no maximal models, Annals of Pure and Applied Logic, vol. 97 (1999), pp. 1–25.Google Scholar
[VD] Vandieren, Monica, Categoricity in abstract elementary classes with no maximal models, Annals of Pure and Applied Logic, to appear, available at http://www.math.lsa.umich.edu/~mvd/home.html.Google Scholar
[Zi] Zilber, Boris, Collected preprints at http://www.maths.ox.ac.uk/~zilber/publ.html. Google Scholar