Andrey Bovykin

Order-types of models of arithmetic and a connection with arithmetic saturation

(Lobachevskii Journal of Mathematics, Vol.16, pp.3 - 15)

First, we study a question we encountered while exploring order-types of models of arithmetic. We prove that if $M \vDash PA$ is resplendent and the lower cofinality of $M \smallsetminus N$ is uncountable then $(M,<)$ is expandable to a model of any consistent theory $T\supseteq PA$ whose set of G\"odel 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 $M \vDash PA$ with $lcf(M\smallsetminus N)>\omega$. The paper concludes with a generalization of theorems of Kossak, Kotlarski and Kaye on automorphisms moving all nondefinable points: a countable model $M\models \PA$ is arithmetically saturated if and only if there is an automorphism $h\colon M\to M$ moving every nondefinable point and such that for all $x\in M$, $Nx$.



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