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Forcing isomorphism II

  • M. C. Laskowski (a1) and S. Shelah (a2) (a3)

Abstract

If T has only countably many complete types, yet has a type of infinite multiplicity then there is a c.c.c. forcing notion such that, in any -generic extension of the universe, there are non-isomorphic models M1 and M2 of T that can be forced isomorphic by a c.c.c. forcing. We give examples showing that the hypothesis on the number of complete types is necessary and what happens if ‘c.c.c’ is replaced by other cardinal-preserving adjectives. We also give an example showing that membership in a pseudo-elementary class can be altered by very simple cardinal-preserving forcings.

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References

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[1]Baldwin, J. T., Fundamentals of stability theory, Springer-Verlag, 1988.
[2]Baldwin, J. T., Laskowski, M. C., and Shelah, S., Forcing isomorphism, this Journal, vol. 58 (1993).
[3]Barwise, J., Back and forth through infinitary logic, Studies in model theory (Morley, M., editor), Mathematical Association of America, 1973, pp. 534.
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[5]Lascar, D., Stability in model theory, Longman, 1987, originally published in French as Stabilité en Théorie des Modèles, 1986.
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[8]Shelah, S., Existence of many L∞,λ-equivalent non-isomorphic models of T of power λ, Annals of Pure and Applied Logic, vol. 34 (1987).
[9]Shelah, S., Classification theory, North-Holland, 1991.
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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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