Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-16T23:47:30.296Z Has data issue: false hasContentIssue false
  • English
  • Français

Examples of non-locality

Published online by Cambridge University Press:  12 March 2014

John T. Baldwin  and
Saharon Shelah
John T. Baldwin
Affiliation:
Department of Mathematics, Statistics and Computer Science M/C 249, University of Illinoisat Chicago, 851 S. Morgan Chicago, Illinois 60607, USA, E-mail: [email protected] Institute of Mathematics, The Hebrew University of Jerusalem, 91904 Jerusalem, Israel
Saharon Shelah
Affiliation:
Institute of Mathematics, The Hebrew University of Jerusalem, 91904 Jerusalem, Israel Department of Mathematics, Rutgers University, New Brunswick, NJ 08854, USA, E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We use κ-free but not Whitehead Abelian groups to construct Abstract Elementary Classes (AEC) which satisfy the amalgamation property but fail various conditions on the locality of Galois-types. We introduce the notion that an AEC admits intersections. We conclude that for AEC which admit intersections, the amalgamation property can have no positive effect on locality: there is a transformation of AEC's which preserves non-locality but takes any AEC which admits intersections to one with amalgamation. More specifically we have: Theorem 5.3. There is an AEC with amalgamation which is not (ℵ0, ℵ1)-tame but is (, ∞)-tame; Theorem 3.3. It is consistent with ZFC that there is an AEC with amalgamation which is not (≤ ℵ2, ≤ ℵ2)-compact.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 73 , Issue 3 , September 2008 , pp. 765 - 782
Copyright
Copyright © Association for Symbolic Logic 2008

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

[Ba100]Baldwin, J. T., Categoricity, Available at www.math.uic.edu/~jbaldwin, 200?Google Scholar
[BCG+00]Baldwin, J., Calvert, W., Goodrick, J., Villaveces, A., and Walczak-Typke, A., Abelian groups as aec's, preprint, 200?Google Scholar
[BK]Baldwin, John T. and Kolesnikov, Alexei, Categoricity, amalgamation, and tameness, preprint:www.math.uic.edu/~jbaldwin, to appear in Israel Journal of Mathematics.Google Scholar
[BKV06]Baldwin, J. T., Kueker, D. W., and VanDieren, M., Upward stability transfer theorem for tame abstract elementary classes, Notre Dame Journal of Formal Logic, vol. 47 (2006), pp. 291–298.CrossRefGoogle Scholar
[EM90]Eklof, P. and Mekler, Alan, Almost free modules: Set theoretic methods, North Holland, 1990.Google Scholar
[GK]Grossberg, Rami and Kolesnikov, Alexei, Excellent abstract elementary classes are tame, preprint.Google Scholar
[Gro02]Grossberg, Rami, Classification theory for non-elementary classes, Logic and algebra (Zhang, Yi, editor), Contemporary Mathematics 302, AMS, 2002, pp. 165–204.CrossRefGoogle Scholar
[GV06a]Grossberg, R. and VanDieren, M., Categoricity from one successor cardinal in tame abstract elementary classes, The Journal of Mathematical Logic, vol. 6 (2006), pp. 181–201.CrossRefGoogle Scholar
[GV06b]Grossberg, R. and VanDieren, M., Galois stability for tame abstract elementary classes, Journal of Mathematical Logic, vol. 6 (2006), pp. 1–24.CrossRefGoogle Scholar
[GV06c]Grossberg, R. and VanDieren, M., Shelah's categoricity conjecture from a successor for tame abstract elementary classes, this Journal, vol. 71 (2006), pp. 553–568.Google Scholar
[HK06]Hyttinen, T. and Kesälä, M., Independence in finitary abstract elementary classes, Annals of Pure and Applied Logic, vol. 143 (2006), no. 1-3, pp. 103–138.CrossRefGoogle Scholar
[Les05]Lessmann, Olivier, Upward categoricity from a successor cardinal for an abstract elementary class with amalgamation, this Journal, vol. 70 (2005), pp. 639–661.Google Scholar
[Rab62]Rabin, Michael, Classes of structures and sets of sentences with the intersection property, Actes du Colloqe de mathématiques a l'Occasion de Tricentenaire de la mort de B. Pascal, Université de Clermont, 1962, pp. 39–53.Google Scholar
[She74]Shelah, S., Infinite abelian groups, Whitehead problem and some constructions, Israel Journal of Mathematics, vol. 18 (1974), pp. 243–256.CrossRefGoogle Scholar
[She87]Shelah, Saharon, Classification of nonelementary classes II, abstract elementary classes, Classification theory (Chicago, IL, 1985) (Baldwin, J. T., editor), Lecture Notes in Mathematics, vol. 1292, Springer, Berlin, 1987, paper 88: Proceedings of the USA–Israel Conference on Classification Theory, Chicago, 12 1985, pp. 419–497.CrossRefGoogle Scholar
[She99]Shelah, S., Categoricity for abstract classes with amalgamation, Annals of Pure and Applied Logic, vol. 98 (1999), pp. 261–294, paper 394. Consult Shelah for post-publication revisions.Google Scholar
[She01]Shelah, S., Categoricity of abstract elementary class in two successive cardinals, Israel Journal of Mathematics, vol. 126 (2001), pp. 29–128, paper 576.CrossRefGoogle Scholar