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Definable types in -minimal theories

  • David Marker (a1) and Charles I. Steinhorn (a2)
Abstract

Let L be a first order language. If M is an L-structure, let LM be the expansion of L obtained by adding constants for the elements of M.

Definition. A type is definable if and only if for any L-formula , there is an LM-formula so that for all iff Mdθ(¯). The formula dθ is called the definition of θ.

Definable types play a central role in stability theory and have also proven useful in the study of models of arithmetic. We also remark that it is well known and easy to see that for MN, the property that every M-type realized in N is definable is equivalent to N being a conservative extension of M, where

Definition. If MN, we say that N is a conservative extension of M if for any n and any LN -definable SNn, SMn is LM-definable in M.

Van den Dries [Dl] studied definable types over real closed fields and proved the following result.

0.1 (van den Dries), (i) Every type over (R, +, -,0,1) is definable.

(ii) Let F and K be real closed fields and F ⊂ K. Then, the following are equivalent:

(a) Every element of K that is bounded in absolute value by an element of F is infinitely close (in the sense of F) to an element of F.

(b) K is a conservative extension of F.

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References
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[D1]van den Dries L. P. D., Tarski's problem and Pfaffian functions, Logic Colloquium '84 (Paris J. B., Wilkie A. J., and Wilmers G. M., editors), North-Holland, Amsterdam, 1986, pp. 5990.
[D2]van den Dries L. P. D., A generalization of the Tarski-Seidenberg theorem, Bulletin of the American Mathematical Society, vol. 15 (1986), pp. 189193.
[K-P-S]Knight J., Pillay A., and Steinhorn C., Definable sets in ordered structures II, Transactions of the American Mathematical Society, vol. 295 (1986), pp. 593605.
[M]Marker D., Omitting types in inline-graphic-minimal theories, this Journal, vol. 51 (1986), pp. 6374.
[P1]Pillay A., Some remarks on definable equivalence relations in inline-graphic-minimal structures, this Journal, vol. 51 (1986), pp. 709714.
[P2]Pillay A., An introduction to stability theory, Oxford University Press, London and New York, 1983.
[P-S1]Pillay A. and Steinhorn C., Definable sets in ordered structures I, Transactions of the American Mathematical Society, vol. 295 (1986), pp. 565592.
[P-S2]Pillay A. and Steinhorn C., Discrete inline-graphic-minimal structures, Annals of Pure and Applied Logic, vol. 34 (1987), pp. 275289.
[P-S3]Pillay A. and Steinhorn C., Definable sets in ordered structures III, Transactions of the American Mathematical Society, vol. 309 (1988), pp. 469476.
[P-S4]Pillay A. and Steinhorn C., Dedekind Complete inline-graphic-minimal structures, this Journal, vol. 52 (1987), pp. 156164.
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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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