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Forcing isomorphism

Published online by Cambridge University Press:  12 March 2014

J. T. Baldwin
Affiliation:
Department of Mathematics, University of Illinois, Chicago, Illinois 60680, E-mail: U128O0@UICVM
M. C. Laskowski
Affiliation:
Department of Mathematics, University of Maryland, College Park, Maryland 20742, E-mail: mcl@math.umd.edu
S. Shelah
Affiliation:
Department of Mathematics, Hebrew University of Jerusalem, Jerusalem, Israel

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.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1993

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References

[1]Baldwin, J. T., Diverse classes, this Journal, vol. 54 (1989), pp. 875893.Google Scholar
[2]Buechler, Steve and Shelah, Saharon, On the existence of regular types, Annals of Pure and Applied Logic, vol. 45 (1989), pp. 207308.Google Scholar
[3]Hart, Bradd, Some results in classification theory, Ph.D. Thesis, McGill University, Montreal, 1986.Google Scholar
[4]Shelah, S., Classification theory and the number of nonisomorphic models, North-Holland, Amsterdam, 1978.Google Scholar
[5]Shelah, S., Classification of first-order theories which have a structure theory, Bulletin of the American Mathematical Society, vol. 12 (1985), pp. 227232.CrossRefGoogle Scholar
[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).CrossRefGoogle Scholar
[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.Google Scholar
[8]Shelah, S., Classification theory, North-Holland, Amsterdam, 1991 (second edition of [4]).Google Scholar
[9]Weiss, W., Versions of Martin's axiom, Handbook of set-theoretic topology (Kunen, K. and Vaughan, J. E., editors), North-Holland, Amsterdam, 1984.Google Scholar
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