Keisler’s order has infinitely many classes
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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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- G. Cherlin and E. Hrushovski, Finite Structures with Few Types, Annals of Mathematics Studies, Vol. 152, Princeton University Press, Princeton, NJ, 2003.Google Scholar
- P. Erdős, A. Hajnal, A.Máté and R. Rado, Combinatorial set Theory: Partition Relations for Cardinals, Studies in Logic and the Foundations of Mathematics, Vol. 106, North-Holland Publishing Co., Amsterdam, 1984.Google Scholar
- E. Hrushovski, Pseudo-finite fields and related structures, in Model Theory and Applications, Quaderni di Matematica, Vol. 11, Aracne, Rome, 2002, pp. 151–212.Google Scholar
- M. Malliaris and S. Shelah, Saturating the random graph with an independent family of small range, in Logic Without Borders, DeGruyter, Berlin–Boston–Munich, 2015.Google Scholar
- S. Mohsenipour and S. Shelah, Set mappings on 4-tuples, paper 1072. Notre Dame Journal of Formal Logic, Accepted.Google Scholar
- M. Morley, 1968 Mathscinet review MR0218224 of Keisler .Google Scholar
- S. Shelah, Classification Theory and the Number of Nonisomorphic Models, Studies in Logic and the Foundations of Mathematics, Vol. 92, North-Holland Publishing Co., Amsterdam, 1990.Google Scholar