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Keisler’s order has infinitely many classes

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Abstract

We prove, in ZFC, that there is an infinite strictly descending chain of classes of theories in Keisler’s order. Thus Keisler’s order is infinite and not a well order. Moreover, this chain occurs within the simple unstable theories, considered model-theoretically tame. Keisler’s order is a central notion of the model theory of the 60s and 70s which compares first-order theories, and implicitly ultrafilters, according to saturation of ultrapowers. Prior to this paper, it was long thought to have finitely many classes, linearly ordered. The model-theoretic complexity we find is witnessed by a very natural class of theories, the n-free k-hypergraphs studied by Hrushovski. This complexity reflects the difficulty of amalgamation and appears orthogonal to forking.

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Authors and Affiliations

  1. Department of Mathematics, University of Chicago, 5734 S. University Avenue, Chicago, IL, 60637, USA

    Maryanthe Malliaris

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

    Saharon Shelah

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

    Saharon Shelah

Authors
  1. Maryanthe Malliaris
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  2. Saharon Shelah
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Correspondence to Maryanthe Malliaris.

Additional information

Malliaris was partially supported by a Sloan research fellowship, by NSF grants DMS-1300634 and DMS-1553653, and by a research membership at MSRI funded through NSF 0932078 000 (Spring 2014).

Shelah was partially supported by European Research Council grant 338821, and NSF grant 1362974 (Rutgers).

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Malliaris, M., Shelah, S. Keisler’s order has infinitely many classes. Isr. J. Math. 224, 189–230 (2018). https://doi.org/10.1007/s11856-018-1647-7

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  • DOI: https://doi.org/10.1007/s11856-018-1647-7

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