Abstract
We apply the recently developed technology of cofinality spectrum problems to prove a range of theorems in model theory. First, we prove that any model of Peano arithmetic is λ-saturated iff it has cofinality ≥ λ and the underlying order has no (κ, κ)-gaps for regular κ < λ. We also answer a question about balanced pairs of models of PA. Second, assuming instances of GCH, we prove that SOP 2 characterizes maximality in the interpretability order ∇*, settling a prior conjecture and proving that SOP 2 is a real dividing line. Third, we establish the beginnings of a structure theory for NSOP 2, proving that NSOP 2 can be characterized by the existence of few so-called higher formulas. In the course of the paper, we show that ps = ts in any weak cofinality spectrum problem closed under exponentiation (naturally defined). We also prove that the local versions of these cardinals need not coincide, even in cofinality spectrum problems arising from Peano arithmetic.
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Malliaris was partially supported by NSF grants 1300634 and 1001666 and by a Sloan fellowship, and by a research membership at MSRI funded by NSF grant 0932078 000 (Spring 2014).
Shelah was partially supported by the Israel Science Foundation grants 710/07 and 1053/11 and European Research Council grant 338821. This is manuscript 1051 in Shelah’s list of publications.
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Malliaris, M., Shelah, S. Model-theoretic applications of cofinality spectrum problems. Isr. J. Math. 220, 947–1014 (2017). https://doi.org/10.1007/s11856-017-1526-7
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DOI: https://doi.org/10.1007/s11856-017-1526-7