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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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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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