Keisler’s order has infinitely many classes
Keisler’s order has infinitely many classes
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凯斯勒阶有无数类
DOI：
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发表时间：
2015
期刊：
影响因子：
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通讯作者：
S. Shelah
中科院分区：
文献类型：
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作者：
M. Malliaris;S. Shelah
我们在 ZFC 中证明了按照凯斯勒顺序存在无限严格下降的理论类别链。因此，凯斯勒的秩序是无限的，而不是一个良好的秩序。此外，这条链发生在简单的不稳定理论中，在理论上被认为是温和的模型。凯斯勒阶数是 60 年代和 70 年代模型理论的核心概念，它根据超幂饱和度对一阶理论和隐式超滤器进行比较。在本文发表之前，长期以来人们一直认为具有有限多个线性排序的类。我们发现的模型理论的复杂性是由一类非常自然的理论证明的，即赫鲁索夫斯基研究的 n 自由 k 超图。这种复杂性反映了合并的难度，并且看起来与分叉正交。
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.