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Forcing isomorphism

  • J. T. Baldwin (a1), M. C. Laskowski (a2) and S. Shelah (a3)

Extract

If two models of a first-order theory are isomorphic, then they remain isomorphic in any forcing extension of the universe of sets. In general however, such a forcing extension may create new isomorphisms. For example, any forcing that collapses cardinals may easily make formerly nonisomorphic models isomorphic. However, if we place restrictions on the partially-ordered set to ensure that the forcing extension preserves certain invariants, then the ability to force nonisomorphic models of some theory T to be isomorphic implies that the invariants are not sufficient to characterize the models of T.

A countable first-order theory is said to be classifiable if it is superstable and does not have either the dimensional order property (DOP) or the omitting types order property (OTOP). If T is not classifiable, Shelah has shown in [5] that sentences in L∞,λ do not characterize models of T of power λ. By contrast, in [8] Shelah showed that if a theory T is classifiable, then each model of cardinality λ is described by a sentence of L∞,λ. In fact, this sentence can be chosen in the . ( is the result of enriching the language by adding for each μ < λ a quantifier saying the dimension of a dependence structure is greater than μ) Further work ([3], [2]) shows that ⊐+ can be replaced by ℵ1.

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References

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[1]Baldwin, J. T., Diverse classes, this Journal, vol. 54 (1989), pp. 875893.
[2]Buechler, Steve and Shelah, Saharon, On the existence of regular types, Annals of Pure and Applied Logic, vol. 45 (1989), pp. 207308.
[3]Hart, Bradd, Some results in classification theory, Ph.D. Thesis, McGill University, Montreal, 1986.
[4]Shelah, S., Classification theory and the number of nonisomorphic models, North-Holland, Amsterdam, 1978.
[5]Shelah, S., Classification of first-order theories which have a structure theory, Bulletin of the American Mathematical Society, vol. 12 (1985), pp. 227232.
[6]Shelah, S., Existence of many L∞,λ-equivalent nonisomorphic models of T of power λ, Annals of Pure and Applied Logic, vol. 34 (1987).
[7]Shelah, S., Universal classes: Part 1, Classification Theory, Chicago 1985 (Baldwin, J., editor), Lecture Notes in Mathematics, vol. 1292, Springer-Verlag, Berlin and New York, 1987, pp. 264419.
[8]Shelah, S., Classification theory, North-Holland, Amsterdam, 1991 (second edition of [4]).
[9]Weiss, W., Versions of Martin's axiom, Handbook of set-theoretic topology (Kunen, K. and Vaughan, J. E., editors), North-Holland, Amsterdam, 1984.
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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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