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Minimal models

  • Rainer Deissler (a1)

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A model is called minimal if it does not contain a proper elementary submodel. A class of models is called Σ1111 resp. elementary) if it is axiomatized by a sentence with σ in and some string of predicate symbols. All languages considered are assumed to be countable. For each model we shall define in a natural way its rank, denoted by rk (), which is an ordinal or ∞. Intuitively speaking, rk () is the least upper bound for the number of steps needed to define the elements of by first order formulas; e.g. we shall have rk((ω, <)) = 1 (each element is f.o. definable), rk ((Z, <)) = 2 (no element is f.o. definable, each element is f.o. definable using any other element as a parameter), rk ((Q, <) ) = ∞ (no element is f.o. definable by any number of steps). This notion of rank leads to a useful game theoretic characterization of minimal models which we apply to show that the Π11 class of minimal models is not Σ11.

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[1]Baldwin, J. T., αT is finite for ℵ1-categorical T, Transactions of the American Mathematical Society, vol. 181 (1973), pp. 3751.
[2]Baldwin, J. T., Blass, A. R., Glass, A. M. and Kueker, D. W., A “natural” theory without a prime model, Algebra Universalis, vol. 3 (1973), pp. 152155.
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[4]Chang, C. C. and Keisler, H. J., Model theory, North-Holland, Amsterdam, 1973.
[5]Deissler, R., Untersuchungen iiber Minimalmodelle, Dissertation, University of Freiburg, 1974.
[6]Flum, J., First-order logic and its extensions, Logic Conference, Kiel, 1974, Springer, Berlin, 1975.
[7]Lopez-Escobar, E. G. K., On defining well ordering, Fundamenta Mathematicae, vol. 59 (1966), pp. 1321.
[8]Vaught, R. L., Descriptive set theory in Lω1ω, Cambridge Summer School in Mathematical Logic, Springer, Berlin, 1973, pp. 574598.

Minimal models

  • Rainer Deissler (a1)

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