Keisler's order has infinitely many classes

M. Malliaris; S. Shelah

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arXiv · 1503.08341

Keisler's order has infinitely many classes

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.

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BibTeX RIS

M. Malliaris, S. Shelah. 2015-03-28. Keisler's order has infinitely many classes. https://arxiv.org/abs/1503.08341

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