### Model-theoretic applications of cofinality spectrum problems

by Malliaris and Shelah. [MiSh:1051]

Israel J Math, 2017

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
lambda-saturated iff it has cofinality >= lambda and the
underlying order has no (kappa, kappa)-gaps for regular kappa
<
lambda . 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
triangleleft^*, 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 {p}_{mathbf {s}} = {t}_{mathbf {s}} in
emph {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.
*2015.02.12 The new note from Feb on the delayed lemma make no
sense:-
the finite satisfiability ensure the formulas are compatible.
Look back at the old.. The second theorem which is delayed (union of
few pairwise non-contradictory) - a debt, look again

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