Andrey Bovykin

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 G√∂del 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, < x < Cl∅ \ ℕ, we have h(x) > x.

 

 


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