Keisler’s order has infinitely many classes

Maryanthe Malliaris*, Saharon Shelah

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

11 Scopus citations

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.

Original languageEnglish
Pages (from-to)189-230
Number of pages42
JournalIsrael Journal of Mathematics
Volume224
Issue number1
DOIs
StatePublished - 1 Apr 2018

Bibliographical note

Publisher Copyright:
© 2018, Hebrew University of Jerusalem.

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