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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
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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.

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

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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