Journal of Symbolic Logic

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

Rami Grossberg and Monica VanDieren

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

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

Theorem. 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:

Corollary. Suppose 𝔎 is a χ-tame abstract elementary class satisfying the amalgamation and joint embedding properties. Let μ₀:= Hanf(𝔎). If χ≤ℶ(2μ₀)⁺ and 𝔎 is categorical in some λ⁺>ℶ(2μ₀)⁺, then 𝔎 is categorical in μ for all μ>ℶ(2μ₀)⁺.

Article information

Source
J. Symbolic Logic, Volume 71, Issue 2 (2006), 553-568.

Dates
First available in Project Euclid: 2 May 2006

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1146620158

Digital Object Identifier
doi:10.2178/jsl/1146620158

Mathematical Reviews number (MathSciNet)
MR2225893

Zentralblatt MATH identifier
1100.03023

Subjects
Primary: 03C45: Classification theory, stability and related concepts [See also 03C48] 03C52: Properties of classes of models 03C75: Other infinitary logic
Secondary: 03C05: Equational classes, universal algebra [See also 08Axx, 08Bxx, 18C05] 03C55: Set-theoretic model theory 03C95: Abstract model theory

Citation

Grossberg, Rami; VanDieren, Monica. Shelah’s categoricity conjecture from a successor for tame abstract elementary classes. J. Symbolic Logic 71 (2006), no. 2, 553--568. doi:10.2178/jsl/1146620158. https://projecteuclid.org/euclid.jsl/1146620158


Export citation

References

  • John Baldwin, Abstract elementary classes, Monograph, in preparation. Available at http://www2.math.uic.edu/~jbaldwin/model.html.
  • --------, Non-splitting extensions, Technical report, available at http://www2.math.uic.edu/~jbaldwin/model.html.
  • John Baldwin, David Kueker, and Monica VanDieren, Upward stability transfer for tame abstract elementary classes, Submitted, http://www.math.lsa.umich.edu/~mvd/home.html.
  • Itay Ben-Yaacov, Positive model theory and compact abstract theories, Journal of Mathematical Logic, vol. 3 (2003), no. 1, pp. 85--118.
  • Alexander Berenstein, Some generalizations of first order tools to homogeneous models, Journal of Symbolic Logic, To appear. Preprint available at http://www.math.uiuc.edu/ãberenst/research.html.
  • Alexander Berenstein and Steven Buechler, A study of independence in strongly homogeneous expansions of Hilbert spaces, Preprint available at http://www.math.uiuc.edu/ ãberenst/research.html.
  • Steven Buechler and Olivier Lessmann, Simple homogeneous models, Journal of the American Mathematical Society, vol. 16 (2003), no. 1, pp. 91--121.
  • Rami Grossberg, Classification theory for non-elementary classes, Logic and algebra (Yi Zhang, editor), Contemporary Mathematics, vol. 302, American Mathematical Society,2002, pp. 165--204.
  • Rami Grossberg and Olivier Lessmann, Shelah's stability spectrum and homogeneity spectrum in finite diagrams, Archives in Mathematical Logic, vol. 41 (2002), no. 1, pp. 1--31.
  • Rami Grossberg and Monica VanDieren, Categoricity from one successor cardinal in tame abstract elementary classes, 17 pages. Submitted. http://www.math.lsa.umich.edu/~mvd/home.html.
  • --------, Galois-stability in tame abstract elementary classes, 23 pages. Submitted in 10/4 /2004. http://www.math.cmu.edu/~rami.
  • Bradd Hart and Saharon Shelah, Categoricity over $P$ for first order $T$ or categoricity for $\phi\in\rm L_ \omega_ 1\omega$ can stop at $\aleph_ k$ while holding for $\aleph_ 0,\dots,\aleph_ k-1$, Israel Journal of Mathematics, vol. 70 (1990), pp. 219--235.
  • C. Ward Henson and José Iovino, 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 C. W. Henson, J. Iovino, A. S. Kechris, and E. W. Odell.
  • Tapani Hyttinen, Generalizing Morley's theorem, Mathematical Logic Quarterly, vol. 44 (1998), pp. 176--184.
  • --------, On nonstructure of elementary submodels of a stable homogeneous structure, Fundamenta Mathematicae, vol. 156 (1998), pp. 167--182.
  • Tapani Hyttinen and Saharon Shelah, Strong splitting in stable homogeneous models, Annals of Pure and Applied Logic, vol. 103 (2000), pp. 201--228.
  • José Iovino, Stable Banach spaces and Banach space structures, I: Fundamentals, Models, algebras, and proofs (X. Caicedo and C. Montenegro, editors), Marcel Dekker, New York,1999, pp. 97--113.
  • --------, Stable Banach spaces and Banach space structures, II: Forking and compact topologies, Models, algebras, and proofs (X. Caicedo and C. Montenegro, editors), Marcel Dekker, New York,1999, pp. 77--95.
  • Bjarni Jónsson, Homogeneous universal systems, Mathematica Scandinavica, vol. 8 (1960), pp. 137--142.
  • H. Jerome Keisler, $L_\omega_1,\omega(\mbox\bfseries\upshape Q)$, Annals of Mathematical Logic, vol. 1 (1969).
  • --------, Model theory for infinitary logic, North-Holland,1971.
  • Oren Kolman and Saharon Shelah, Categoricity of Theories in $L_\kappa, \omega$ when $\kappa$ is a measurable cardinal. Part I, Fundamentae Mathematicae, vol. 151 (1996), pp. 209--240.
  • Olivier Lessmann, Pregeometries in finite diagrams, Annals of Pure and Applied Logic, vol. 106 (2000), no. 1--3, pp. 49--83.
  • --------, Upward categoricity from a successor cardinal for tame abstract classes with amalgamation, Journal of Symbolic Logic, vol. 70 (2005), no. 2, pp. 639--660.
  • Jerzy Łoś, On the categoricity in power of elementary deductive systems and related problems, Colloquium Mathematicum, vol. 3 (1954), pp. 58--62.
  • Michael Makkai and Saharon Shelah, Categoricity of theories $L_\kappa\omega$ with $\kappa$ a compact cardinal, Annals of Pure and Applied Logic, vol. 47 (1990), pp. 41--97.
  • Leo Marcus, A prime minimal model with an infinite set of indiscernibles, Israel Journal of Mathematics, vol. 11 (1972), pp. 180--183.
  • Michael Morley, Categoricity in power, Transactions of the American Mathematical Society, vol. 114 (1965), pp. 514--538.
  • Saharon Shelah, Finite diagrams stable in power, Annals of Mathematical Logic, vol. 2 (1970), pp. 69--118.
  • --------, Categoricity of uncountable theories, Proceedings of the Tarski Symposium (L. A. Henkin et al., editors), AMS, Providence, R.I.,1974, pp. 187--203.
  • --------, Categoricity in $\aleph_1$ of sentences in $L_\omega_1,\omega(Q)$, Israel Journal of Mathematics, (1975), pp. 127--148.
  • --------, The lazy model-theoretician's guide to stability, Logique et Analyse, vol. 18 (1975), pp. 241--308.
  • --------, Classification theory for nonelementary classes, I. The number of uncountable models of $\psi \in L_\omega _1,\omega $. Part A, Israel Journal of Mathematics, vol. 46 (1983), pp. 212--240.
  • --------, Classification theory for nonelementary classes, I. The number of uncountable models of $\psi \in L_\omega _1,\omega $. Part B, Israel Journal of Mathematics, vol. 46 (1983), pp. 241--273.
  • --------, Classification of nonelementary classes. II. Abstract elementary classes, Classification theory, Lecture Notes in Mathematics, vol. 1292, Springer-Berlin,1987, pp. 419--497.
  • --------, Classification theory and the number of non-isomorphic models, 2 ed., North Holland Amsterdam,1990.
  • --------, Categoricity of abstract classes with amalgamation, Annals of Pure and Applied Logic, vol. 98 (1999), no. 1--3, pp. 241--294.
  • --------, On what I do not understand (and have something to say), model theory, Mathematica Japonica, vol. 51 (2000), pp. 329--377.
  • --------, Categoricity of an abstract elementary class in two successive cardinals, Israel Journal of Mathematics, vol. 126 (2001), pp. 29--128.
  • --------, Categoricity of theories in $L_\kappa^*,\omega$ when $\kappa^*$ is a measurable cardinal. Part II. Dedicated to the memory of Jerzy Łoś, Fundamenta Mathematica, vol. 170 (2001), no. 1--2, pp. 165--196.
  • --------, Categoricity in abstract elementary classes: going up inductive step,Preprint, 100 pages.
  • --------, Toward classification theory of good $\lambda$ frames and abstract elementary classes.
  • Saharon Shelah and Andrés Villaveces, Categoricity in abstract elementary classes with no maximal models, Annals of Pure and Applied Logic, vol. 97 (1999), no. 1--3, pp. 1--25.
  • Monica VanDieren, Categoricity in abstract elementary classes with no maximal models, Annals of Pure and Applied Logic, 61 pages, accepted, http://www.math.lsa.umich.edu/~mvd/home.html.
  • Boris Zilber, Analytic and pseudo-analytic structures, Preprint, http://www.maths.ox.ac.uk/~zilber.