TY - JOUR
T1 - Keisler’s order has infinitely many classes
AU - Malliaris, Maryanthe
AU - Shelah, Saharon
N1 - Publisher Copyright:
© 2018, Hebrew University of Jerusalem.
PY - 2018/4/1
Y1 - 2018/4/1
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=85044241231&partnerID=8YFLogxK
U2 - 10.1007/s11856-018-1647-7
DO - 10.1007/s11856-018-1647-7
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AN - SCOPUS:85044241231
SN - 0021-2172
VL - 224
SP - 189
EP - 230
JO - Israel Journal of Mathematics
JF - Israel Journal of Mathematics
IS - 1
ER -