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

A. Bovykin

Lobachevskii journal of mathematics, Tome 16 (2004), pp. 3-15

Résumé

First, we study a question we encountered while exploring order-types of models of arithmetic. We prove that if $M\vDash{\rm PA}$ is resplendent and the lower cofinality of $M\setminus\mathbb N$ is uncountable then $(M,)$ is expandable to a model of anyconsistent theory $T\supseteq{\rm PA}$ whose set of Göodel 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 $M\vDash{\rm PA}$ with lcf $(M\setminus\mathbb N)>\omega$. The paper concludes with a generalization of theorems of Kossak, Kotlarski and Kaye on automorphisms moving all nondefinable points: a countable model $M\vDash{\rm PA}$ 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 $x\in M$, $\mathbb N$, we have $h(x)>x$.

Détail

Keywords: models of Peano arithmetic, linearly ordered sets, arithmetic saturation, resplendency, automorphisms of models.