Order-types of models of arithmetic and a connection with arithmetic saturation
(Lobachevskii Journal of Mathematics, Volume XVI)
First, we study a question we encountered while exploring
order-types of models of arithmetic. We prove that if M
PA is resplendent
and the lower cofinality of M \ ℕ is uncountable then (M,<) is expandable
to a model of any consistent theory T ⊇ PA whose set of Gödel numbers is
arithmetic. This leads to the following characterization of Scott sets closed
under jump: a Scott set X is closed under jump if and only if X is the set of
all sets of natural numbers definable in some recursively saturated model
M
PA with lcf(M \ ℕ) > ω. The paper concludes with a generalization of
theorems of Kossak, Kotlarski and Kaye on automorphisms moving all
nondefinable points: a countable model M
PA is arithmetically saturated
if and only if there is an automorphism h: M → M moving every
nondefinable point and such that for all x 
M, ℕ < x < Cl∅ \ ℕ, we have
h(x) > x.