Hostname: page-component-74656f9884-j7cmm Total loading time: 0 Render date: 2026-11-10T21:24:05.263Z Has data issue: false hasContentIssue false
  • English
  • Français

The amalgamation spectrum

Published online by Cambridge University Press:  12 March 2014

John T. Baldwin ,
Alexei Kolesnikov  and
Saharon Shelah
John T. Baldwin
Affiliation:
Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, 851 S. Morgan Street, Chicago, Il 60607, USA, E-mail: jbaldwin@uic.edu
Alexei Kolesnikov
Affiliation:
Towson University, Department of Mathematics, Towson, Md 21252, USA, E-mail: AKolesnikov@towson.edu
Saharon Shelah
Affiliation:
The Hebrew University of JerusalemJerusalem 91904, Israel Department of Mathematics, Rutgers University, New Brunswick, Nj 08854, USA, E-mail: shlhetal@math.huji.ac.il
Get access Rights & Permissions [Opens in a new window]

Abstract

We study when classes can have the disjoint amalgamation property for a proper initial segment of cardinals.

For every natural number k, there is a class Kk, defined by a sentence in Lω1,ω that has no models of cardinality greater than ℶk + 1, but Kk has the disjoint amalgamation property on models of cardinality less than or equal to ℵk − 3 and has models of cardinality ℵk − 1.

More strongly, we can have disjoint amalgamation up to ℵ∝ for ∝ < ω1, but have a bound on size of models.

For every countable ordinal ∝, there is a class K∝ defined by a sentence in Lω1,ω that has no models of cardinality greater than ℶω1, but K does have the disjoint amalgamation property on models of cardinality less than or equal to ℵ∝.

Finally we show that we can extend the ℵ∝ to ℶ∝ in the second theorem consistently with ZFC and while having ℵi ≪ ℶi for 0 < i < ∝. Similar results hold for arbitrary ordinals ∝ with ∣∝∣ = k and Lk + ω.

Information

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 74 , Issue 3 , September 2009 , pp. 914 - 928
Copyright
Copyright © Association for Symbolic Logic 2009

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

[Bal]Baldwin, John T., Categoricity, www.math.uic.edu/~jbaldwin.Google Scholar
[Gro02]Grossberg, Rami, Classification theory for non-elementary classes, Logic and algebra (Zhang, Yi, editor), Contemporary Mathematics, vol. 302, American Mathematical Society, 2002, pp. 165–204.CrossRefGoogle Scholar
[GV06]Grossberg, Rami and Dieren, M. Van, Shelah's categoricity conjecture from a successor for tame abstract elementary classes, this Journal, vol. 71 (2006), pp. 553–568.Google Scholar
[Hjo07]Hjorth, Greg, Knight's model, its automorphism group, and characterizing the uncountable cardinals, Notre Dame Journal of Formal Logic, (2007).Google Scholar
[Kni77]Knight, J.F., A complete l ω1, ω-sentence characterizing ℵ1, this Journal, vol. 42 (1977), pp. 151–161.Google Scholar
[LS93]Laskowski, Michael C. and Shelah, Saharon, On the existence of atomic models, this Journal, vol. 58 (1993), pp. 1189–1194.Google Scholar
[Les]Lessmann, Olivier, Upward categoricity from a successor cardinal for an abstract elementary class with amalgamation, this Journal, vol. 70 (2005), no. 2, pp. 639–660.Google Scholar
[Mor65]Morley, M., Omitting classes of elements, The theory of models (Addison, , Henkin, , and Tarski, , editors), North-Holland, Amsterdam, 1965, pp. 265–273.Google Scholar
[She78a]Shelah, Saharon, Classification theory and the number of nonisomorphic models, North-Holland, 1978.Google Scholar
[She78b]Shelah, Saharon, A weak generalization of M A to higher cardinals, Israel Journal of Mathematics, vol. 30 (1978), pp. 297–306.CrossRefGoogle Scholar
[She80]Shelah, Saharon, Simple unstable theories, Annals of Mathematical Logic, vol. 19 (1980), pp. 177–203.CrossRefGoogle Scholar
[She83a]Shelah, Saharon, Classification theory for nonelementary classes. I. The number of uncountable models of ψ ∈ L ω1ω, part A, Israel Journal of Mathematics, vol. 46 (1983), no. 3, pp. 212–240, paper 87a.CrossRefGoogle Scholar
[She83b]Shelah, Saharon, Classification theory for nonelementary classes. I. The number of uncountable models of ψ ∈ L ω1ω part B, Israel Journal of Mathematics, vol. 46 (1983), no. 3, pp. 241–271, paper 87b.CrossRefGoogle Scholar
[She99]Shelah, Saharon, 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.CrossRefGoogle Scholar
[Sou]Souldatos, Ioannis, Notes on cardinals that characterizable by a complete (Scott) sentence, preprint.Google Scholar