arXiv 1503.083412015-03math.LO

Keisler's order has infinitely many classes

M. Malliaris, S. Shelah

备注：: Minor edits, updated cross-references
发表信息：: Israel J. Math. 224 No. 1 (2018) 189--230

英文摘要：

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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