Hostname: page-component-8448b6f56d-tj2md Total loading time: 0 Render date: 2024-04-16T07:05:52.469Z Has data issue: false hasContentIssue false
  • English
  • Français

Shelah's categoricity conjecture from a successor for tame abstract elementary classes

Published online by Cambridge University Press:  12 March 2014

Rami Grossberg  and
Monica Vandieren
Rami Grossberg
Affiliation:
Department of Mathematics, Carnegie Mellon University, Pittsburgh
Monica Vandieren
Affiliation:
Department of Mathematics, University of Michigan
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove a categoricity transfer theorem for tame abstract elementary classes.

Suppose that K is a χ-tame abstract elementary class and satisfies the amalgamation and joint embedding properties and has arbitrarily large models. Let λ ≥ Max{χ, LS(K+}. If K is categorical in λ and λ+, then K is categorical in λ++.

Combining this theorem with some results from [37]. we derive a form of Shelah's Categoricity Conjecture for tame abstract elementary classes:

Suppose K is χ-tame abstract elementary class satisfying the amalgamation and joint embedding properties. Let μ0 ≔ Hanf(K). Ifand K is categorical in somethen K is categorical in μ for all μ .

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 71 , Issue 2 , June 2006 , pp. 553 - 568
Copyright
Copyright © Association for Symbolic Logic 2006

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]Baldwin, John, Abstract elementary classes, Monograph, in preparation. Available at http://www2.math.uic.edu/~jbaldwin/model.html.Google Scholar
[2]Baldwin, John, Non-splitting extensions. Technical report, available at http://www2.math.uic.edu/~jbaldwin/model.html.Google Scholar
[3]Baldwin, John, Kueker, David, and VanDieren, Monica, Upward stability transfer for tame abstract elementary classes. Submitted, http://www.math.lsa.umich.edu/~mvd/home.html.Google Scholar
[4]Ben-Yaacov, Itay, Positive model theory and compact abstract theories, Journal of Mathematical Logic, vol. 3 (2003), no. 1, pp. 85–118.CrossRefGoogle Scholar
[5]Berenstein, Alexander, Some generalizations of first order tools to homogeneous models, this Journal. To appear. Preprint available at http://www.math.uiuc.edu/~aberenst/research.html.Google Scholar
[6]Berenstein, Alexander and Buechler, Steven, A study of independence in strongly homogeneous expansions of Hilbert spaces. Preprint available at http://www.math.uiuc.edu/~aberenst/research.html.Google Scholar
[7]Buechler, Steven and Lessmann, Olivier, Simple homogeneous models, Journal of the American Mathematical Society, vol. 16 (2003), no. 1. pp. 91–121.CrossRefGoogle Scholar
[8]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
[9]Grossberg, Rami and Lessmann, Olivier, Shelah's stability spectrum and homogeneity spectrum in finite diagrams. Archives in Mathematical Logic, vol. 41 (2002), no. 1. pp. 1–31.CrossRefGoogle Scholar
[10]Grossberg, Rami and VanDieren, Monica, Categoricity from one successor cardinal in tame abstract elementary classes, 17 pages. Submitted, http://www.math.lsa.umich.edu/~mvd/home.html.Google Scholar
[11]Grossberg, Rami and VanDieren, Monica, Galois-stability in tame abstract elementary classes. 23 pages. Submitted in 10/4/2004. http://www.math.cmu.edu/~rami.Google Scholar
[12]Hart, Bradd and Shelah, Saharon, Categoricity over P for first order T or categoricity for ϕ ϵ Lω1ω can stop at ℵk while holding for ℵ0, …, ℵk−1. Israel Journal of Mathematics, vol. 70 (1990). pp. 219–235.Google Scholar
[13]Henson, C. Ward and Iovino, José, Ultraproducts in analysis. Analysis and Logic, London Mathematical Society Lecture Note Series, Cambridge University Press, to appear. Part I of the three part book by Henson, C. W., Iovino, J., Kechris, A. S., and Odell, E. W..CrossRefGoogle Scholar
[14]Hyttinen, Tapani, Generalizing Morley's theorem, Mathematical Logic Quarterly, vol. 44 (1998), pp. 176–184.CrossRefGoogle Scholar
[15]Hyttinen, Tapani, On nonstructure of elementary submodels of a stable homogeneous structure, Fundamenta Mathematical vol. 156 (1998), pp. 167–182.CrossRefGoogle Scholar
[16]Hyttinen, Tapani and Shelah, Saharon, Strong splitting in stable homogeneous models, Annals of Pure and Applied Logic, vol. 103 (2000), pp. 201–228.CrossRefGoogle Scholar
[17]Iovino, José, Stable Banach spaces and Banach space structures, I: Fundamentals, Models, algebras, and proofs (Caicedo, X. and Montenegro, C., editors). Marcel Dekker, New York, 1999, pp. 97–113.Google Scholar
[18]Iovino, José, Stable Banach spaces and Banach space structures, II: Forking and compact topologies, Models, algebras, and proofs (Caicedo, X. and Montenegro, C., editors). Marcel Dekker, New York, 1999. pp. 77–95.Google Scholar
[19]Jónsson, Bjarni, Homogeneous universal systems, Mathematica Scandinavica, vol. 8 (1960), pp. 137–142.CrossRefGoogle Scholar
[20]Keisler, H. Jerome, Lω1ω(Q). Annals of Mathematical Logic, vol. 1 (1969).Google Scholar
[21]Keisler, H. Jerome, Model theory for infinitary logic, North-Holland, 1971.Google Scholar
[22]Kolman, Oren and Shelah, Saharon, Categoricity of Theories in Lk,ω when κ is a measurable cardinal, Part I. Fundamentae Mathematicae, vol. 151 (1996), pp. 209–240.Google Scholar
[23]Lessmann, Olivier, Pregeometries infinite diagrams, Annals of Pure and Applied Logic, vol. 106, (2000). no. 1–3. pp. 49–83.CrossRefGoogle Scholar
[24]Lessmann, Olivier, Upward categoricity from a successor cardinal for tame abstract classes with amalgamation, this Journal, vol. 70 (2005). no. 2, pp. 639–660.Google Scholar
[25]Łoś, Jerzy, On the categoricity in power of elementary deductive systems and related problems, Colloquium Mathematicum, vol. 3 (1954), pp. 58–62.CrossRefGoogle Scholar
[26]Makkai, Michael and Shelah, Saharon, Categoricity of theories Lkω with κ a compact cardinal, Annals of Pure and Applied Logic, vol. 47 (1990), pp. 41–97.Google Scholar
[27]Marcus, Leo, A prime minimal model with an infinite set of indiscernibles, Israel Journal of Mathematics, vol. 11 (1972), pp. 180–183.CrossRefGoogle Scholar
[28]Morley, Michael, Categoricity in power, Transactions of the American Mathematical Society. vol. 114 (1965). pp. 514–538.CrossRefGoogle Scholar
[29]Shelah, Saharon, Finite diagrams stable in power, Annals of Mathematical Logic, vol. 2 (1970), pp. 69–118.CrossRefGoogle Scholar
[30]Shelah, Saharon, Categoricity of uncountable theories, Proceedings of the Tarski Symposium (Henkin, L. A.et al., editors), AMS, Providence, R.I., 1974, pp. 187–203.CrossRefGoogle Scholar
[31]Shelah, Saharon, Categoricity in ℵ1 of sentences in (Q). Israel Journal of Mathematics, (1975). pp. 127–148.CrossRefGoogle Scholar
[32]Shelah, SaharonThe lazy model-theoretician's guide to stability, Logique et Analyse, vol. 18 (1975), pp. 241–308.Google Scholar
[33]Shelah, Saharon, Classification theory for nonelementary classes. I. The number of uncountable models of Ψ ϵ Lω1ω, Part A, Israel Journal of Mathematics, vol. 46 (1983). pp. 212–240.CrossRefGoogle Scholar
[34]Shelah, Saharon, Classification theory for nonelementary classes. I. The number of uncountable models of Ψ ϵ Lω1ω. Part B. Israel Journal of Mathematics, vol. 46 (1983). pp. 241–273.CrossRefGoogle Scholar
[35]Shelah, Saharon, Classification of nonelementary classes. II. Abstract elementary classes, Classification theory, Lecture Notes in Mathematics, vol. 1292. Springer-Berlin, 1987, pp. 419–497.CrossRefGoogle Scholar
[36]Shelah, Saharon, Classification theory and the number of non-isomorphic models, 2 ed., North Holland, Amsterdam, 1990.Google Scholar
[37]Shelah, Saharon, Categoricity of abstract classes with amalgamation, Annals of Pure and Applied Logic, vol. 98 (1999). no. 1–3, pp. 241–294.CrossRefGoogle Scholar
[38]Shelah, Saharon, On what I do not understand (and have something to say), model theory, Mathematica Japonica, vol. 51 (2000), pp. 329–377.Google Scholar
[39]Shelah, Saharon, Categoricity of an abstract elementary class in two successive cardinals, Israel Journal of Mathematics, vol. 126 (2001), pp. 29–128.CrossRefGoogle Scholar
[40]Shelah, Saharon, Categoricity of theories in Lκ*ω when κ* is a measurable cardinal. Part II. Dedicated to the memory of Jerzy Łoś, Fundamenta Mathematica, vol. 170 (2001). no. 1–2, pp. 165–196.CrossRefGoogle Scholar
[41]Shelah, Saharon, Categoricity in abstract elementary classes: going up inductive step. Preprint. 100 pages.Google Scholar
[42]Shelah, Saharon, Toward classification theory of good λ frames and abstract elementary classes.Google Scholar
[43]Shelah, Saharon and Villaveces, Andrés. Categoricity in abstract elementary classes with no maximal models, Annals of Pure and Applied Logic, vol. 97 (1999). no. 1–3. pp. 1–25.CrossRefGoogle Scholar
[44]VanDieren, Monica, Categoricity in abstract elementary classes with no maximal models, Annals of Pure and Applied Logic. 61 pages, accepted, http://www.math.Isa.umich.edu/~mvd/home.html.Google Scholar
[45]Zilber, Boris, Analytic and pseudo-analytic structures, Preprint, http://www.maths.ox.ac.uk/~zilber.Google Scholar