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Strongly dependent theories

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Abstract

We further investigate the class of models of a strongly dependent (first order complete) theory T, continuing [Sh:715], [Sh:783] and related works. Those are properties (= classes) somewhat parallel to superstability among stable theory, though are different from it even for stable theories. We show equivalence of some of their definitions, investigate relevant ranks and give some examples, e.g., the first order theory of the p-adics is strongly dependent. The most notable result is: if |A| + |T| ≤ µ, I ⊆ ℭ and |I|≥ℶ|T|+(µ), then some J ⊆ I of cardinality µ+ is an indiscernible sequence over A.

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References

  1. J. T. Baldwin and J. Saxl, Logical stability in group theory, Journal of the Australian Mathematical Society, Ser. A 21 (1976), 267–276.

    Article  MathSciNet  MATH  Google Scholar 

  2. G. Cherlin, Superstable division rings, in Logic Colloquium’ 77 (Proc. Conf., Wrocław, 1977), Studies in Logic and the Foundations of Mathematics, Vol. 96, North-Holland, Amsterdam-Newyork, 1978, pp. 99–111.

    Chapter  Google Scholar 

  3. G. Cherlin and S. Shelah, Superstable fields and groups, Annals of Mathematical Logic 18 (1980), 227–270.

    Article  MathSciNet  Google Scholar 

  4. R. Cluckers, L. Lipshitz and Z. Robinson, Analytic cell decomposition and analytic motivic integration, Annales Scientifiques de l’École Normale Supérieure, to appear. math.AG/0503722.

  5. M. Cohen and S. Shelah, Stable theories and representation over sets, Mathematical Logic Quarterly, submitted. 0906.3050.

  6. M. Cohen and S. Shelah, Ranks for strongly dependent theories, preprint. math.LO/1303.3441.

  7. E. Firstenberg and S. Shelah, Perpendicular indiscernible sequences in real closed fields, preprint. math.LO/1208.1302.

  8. Y. Gurevich, Monadic theory of order and topology. I., Israel Journal of Mathematics 27 (1977), 299–319.

    Article  MathSciNet  MATH  Google Scholar 

  9. I. Kaplan and S. Shelah, A counterexample to existence of indiscernibles and other examples, Journal of Symbolic Logic, accepted. math.LO/1009.5420.

  10. I. Kaplan and S. Shelah, A dependent theory with few indiscernibles, Israel Journal of Mathematics, accepted. math.LO/1010.0388.

  11. I. Kaplan and S. Shelah, Chain conditions in dependent groups, Annals of Pure and Applied Logic 164 (2013), 1322–1337.

    Article  MathSciNet  Google Scholar 

  12. A. Macintyre, On ω 1-categorical theories of fields, Fundamenta Mathematicae 71 (1971), 1–25.

    MathSciNet  MATH  Google Scholar 

  13. A. Onshuus, th-Forking, algebraic independence and examples of Rosy theories, Preprint (2003). math.LO/0306003.

  14. A. Onshuus, Properties and consequences of th-independence, Preprint (2004). math.LO/0205004.

  15. J. Pas, Cell decomposition and local zeta functions in a tower of unramified extensions of a p-adic field, Proceedings of the London Mathematical Society 60 (1990), 37–67.

    Article  MathSciNet  MATH  Google Scholar 

  16. S. Shelah, A dependent dream and recounting types, preprint. math.LO/1202.5795.

  17. S. Shelah, Definable groups for dependent and 2-dependent theories, Journal of Symbolic Logic, submitted. math.LO/0703045.

  18. S. Shelah, Dependent T and existence of limit models, Tbilisi Mathematical Journal, submitted. math.LO/0609636.

  19. S. Shelah, Dependent theories and the generic pair conjecture, Communications in Contemporary Mathematics, accepted. math.LO/0702292.

  20. S. Shelah, No limit model in inaccessibles, CRM Proceedings and Lecture Notes 53 (2011), 277–290. math.LO/0705.4131.

    Google Scholar 

  21. S. Shelah, Stable frames and weights, preprint.

  22. S. Shelah, Theories with EF-equivalent non-isomorphic models II, preprint.

  23. S. Shelah, Simple unstable theories, Annals of Mathematical Logic 19 (1980), 177–203.

    Article  MathSciNet  MATH  Google Scholar 

  24. S. Shelah, Monadic logic: Hanf numbers, in Around Classification Theory of Models, Volume 1182 of Lecture Notes in Mathematics, Springer, Berlin, 1986, pp. 203–223.

    Chapter  Google Scholar 

  25. S. Shelah, Classification Theory and the Number of Nonisomorphic Models, Volume 92 of Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 1990, xxxiv+705 pp.

    Google Scholar 

  26. S. Shelah, On what I do not understand (and have something to say), model theory, Mathematica Japonica 51 (2000), 329–377. math.LO/9910158.

    MathSciNet  MATH  Google Scholar 

  27. S. Shelah, Classification theory for elementary classes with the dependence property — a modest beginning, Scientiae Mathematicae Japonicae 59, No. 2 (special issue: e9, 503–544), (2004), 265–316. math.LO/0009056.

    Google Scholar 

  28. S. Shelah, Quite complete real closed fields, Israel Journal of Mathematics 142 (2004), 261–272. math.LO/0112212.

    Article  MathSciNet  MATH  Google Scholar 

  29. S. Shelah, Minimal bounded index subgroup for dependent theories, Proceedings of the American Mathematical Society 136 (2008), 1087–1091. math.LO/0603652.

    Article  MathSciNet  MATH  Google Scholar 

  30. S. Shelah, Theories with EF-equivalent non-isomorphic models, Tbilisi Mathematical Journal 1 (2008), 133–164. math.LO/0703477.

    MathSciNet  MATH  Google Scholar 

  31. S. Shelah, Dependent first order theories, continued, Israel Journal of Mathematic 173 (2009). math.LO/0406440.

  32. S. Shelah, Quite complete real closed fields revisited, preprint.

  33. S. Shelah, Representation over orders of elementary classes, preprint.

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Authors and Affiliations

  1. Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Edmond J. Safra Campus, Givat Ram, Jerusalem, 91904, Israel

    Saharon Shelah

  2. Department of Mathematics, Hill Center-Busch Campus, Rutgers, The State University of New Jersey, 110 Frelinghuysen Road, Piscataway, NJ, 08854-8019, USA

    Saharon Shelah

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  1. Saharon Shelah
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Correspondence to Saharon Shelah.

Additional information

This research was supported by the Israel Science Foundation. Publication 863.

I would like to thank Alice Leonhardt for the beautiful typing.

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Shelah, S. Strongly dependent theories. Isr. J. Math. 204, 1–83 (2014). https://doi.org/10.1007/s11856-014-1111-2

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  • DOI: https://doi.org/10.1007/s11856-014-1111-2

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