TY - JOUR
T1 - Keisler's order is not simple (and simple theories may not be either)
AU - Malliaris, M.
AU - Shelah, S.
N1 - Publisher Copyright:
© 2021 Elsevier Inc.
PY - 2021/12/3
Y1 - 2021/12/3
N2 - Solving a decades-old problem we show that Keisler's 1967 order on theories has the maximum number of classes. In fact, it embeds P(ω)/fin. The theories we build are simple unstable with no nontrivial forking, and reflect growth rates of sequences which may be thought of as densities of certain regular pairs, in the sense of Szemerédi's regularity lemma. The proof involves ideas from model theory, set theory, and finite combinatorics.
AB - Solving a decades-old problem we show that Keisler's 1967 order on theories has the maximum number of classes. In fact, it embeds P(ω)/fin. The theories we build are simple unstable with no nontrivial forking, and reflect growth rates of sequences which may be thought of as densities of certain regular pairs, in the sense of Szemerédi's regularity lemma. The proof involves ideas from model theory, set theory, and finite combinatorics.
KW - Keisler's order
KW - Model theory
KW - Regular ultrafilters
KW - Simple theories
UR - http://www.scopus.com/inward/record.url?scp=85115349910&partnerID=8YFLogxK
U2 - 10.1016/j.aim.2021.108036
DO - 10.1016/j.aim.2021.108036
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AN - SCOPUS:85115349910
SN - 0001-8708
VL - 392
JO - Advances in Mathematics
JF - Advances in Mathematics
M1 - 108036
ER -