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Mathematics > Logic

arXiv:1503.08341 (math)
[Submitted on 28 Mar 2015 (v1), last revised 20 Aug 2015 (this version, v2)]

Title:Keisler's order has infinitely many classes

Authors:M. Malliaris, S. Shelah
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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.
Comments: Minor edits, updated cross-references
Subjects: Logic (math.LO)
Report number: MiSh:1050
Cite as: arXiv:1503.08341 [math.LO]
  (or arXiv:1503.08341v2 [math.LO] for this version)
  https://doi.org/10.48550/arXiv.1503.08341

Submission history

From: Maryanthe Malliaris [view email]
[v1] Sat, 28 Mar 2015 19:59:20 UTC (131 KB)
[v2] Thu, 20 Aug 2015 10:36:33 UTC (125 KB)
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