Abstract.
We show in the paper that for any non-classifiable countable theory T there are non-isomorphic models and that can be forced to be isomorphic without adding subsets of small cardinality. By making suitable cardinal arithmetic assumptions we can often preserve stationary sets as well. We also study non-structure theorems relative to the Ehrenfeucht-Fraïssé game.
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References
Abraham, U., Shelah, S.: Forcing closed unbounded sets. J. Symbolic Logic 48, 643–657 (1983)
Baldwin, J.T., Laskowski, M.C., Shelah, S.: Forcing isomorphism. J. Symbolic Logic 58, 1291–1301 (1993)
Baumgartner, J.E., Harrington, L.A., Kleinberg, E.M.: Adding a closed unbounded set. J. Symbolic Logic 41, 481–482 (1976)
Huuskonen, T., Hyttinen, T., Rautila, M.: On the κ-cub game on λ and I[λ]. Arch. Math. Logic 38, 549–557 (1999)
Hyttinen, T., Shelah, S.: Constructing strongly equivalent nonisomorphic models for unsuperstable theories. Part A. J. Symbolic Logic 59, 984–996 (1994)
Hyttinen, T., Shelah, S.: Constructing strongly equivalent nonisomorphic models for unsuperstable theories. Part B. J. Symbolic Logic 60, 1260–1272 (1995)
Hyttinen, T., Shelah, S.: Constructing strongly equivalent nonisomorphic models for unsuperstable theories. Part C. J. Symbolic Logic 64, 634–642 (1999)
Hyttinen, T., Shelah, S., Tuuri, H.: Remarks on strong nonstructure theorems. Notre Dame J. Formal Logic 34, 157–168 (1993)
Hyttinen, T., Tuuri, H.: Constructing strongly equivalent nonisomorphic models for unstable theories. Ann. Pure Appl. Logic 52, 203–248 (1991)
Laskowski, M.C., Shelah, S.: Forcing isomorphism. II. J. Symbolic Logic 61, 1305–1320 (1996)
Mitchell, W.: Aronszajn trees and the independence of the transfer property. Ann. Math. Logic 5, 21–46 (1972)
Nadel, M., Stavi, J.: L ∞ λ -equivalence, isomorphism and potential isomorphism. Trans. Amer. Math. Soc. 236, 51–74 (1978)
Shelah, S.: Non structure theory. In preparation
Shelah, S.: Classification theory and the number of nonisomorphic models. North-Holland Publishing Co., Amsterdam, second edition, 1990
Shelah, S.: The number of non-isomorphic models of an unstable first-order theory. Israel J. Math. 9, 473–487 (1971)
Shelah, S.: Classification of nonelementary classes. II. Abstract elementary classes. In J. T. Baldwin, editor, Proceedings of the USA–Israel Conference on Classification Theory, volume 1292, Lecture Notes in Mathematics. Springer-Verlag, 1987
Shelah, S.: Existence of many L ∞,λ-equivalent, nonisomorphic models of T of power λ. Ann. Pure Appl. Logic 34, 291–310 (1987). Stability in model theory (Trento, 1984)
Shelah, S.: There are Jonsson algebras in many inaccessible cardinals. In: Cardinal Arithmetic, volume~29, Oxford Logic Guides. Oxford University Press, 1994
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The research of the first and second author was partially supported by Academy of Finland grant 40734
Mathematics Subject Classification (2000): 03C55, 03C45
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Huuskonen, T., Hyttinen, T. & Rautila, M. On potential isomorphism and non-structure. Arch. Math. Logic 43, 85–120 (2004). https://doi.org/10.1007/s00153-003-0185-z
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DOI: https://doi.org/10.1007/s00153-003-0185-z