Israel Journal of Mathematics

, Volume 224, Issue 1, pp 189–230 | Cite as

Keisler’s order has infinitely many classes

  • Maryanthe Malliaris
  • Saharon Shelah


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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© Hebrew University of Jerusalem 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ChicagoChicagoUSA
  2. 2.Einstein Institute of Mathematics, Edmond J. Safra CampusThe Hebrew University of JerusalemGivat Ram, JerusalemIsrael
  3. 3.Department of Mathematics, Hill Center - Busch Campus, RutgersThe State University of New JerseyPiscatawayUSA

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