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Lobachevskii Journal of Mathematics, 2004, том 16, страницы 3–15
(Mi ljm81)
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Order-types of models of arithmetic and a connection with arithmetic
saturation
A. Bovykin St. Petersburg Department of V. A. Steklov Institute of Mathematics, Russian Academy of Sciences
Аннотация:
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 $x\in M$, $\mathbb N<x<{\rm Cl}\oplus\setminus\mathbb N$, we have $h(x)>x$.
Ключевые слова:
models of Peano arithmetic, linearly ordered sets, arithmetic saturation, resplendency, automorphisms of models.
Образец цитирования:
A. Bovykin, “Order-types of models of arithmetic and a connection with arithmetic
saturation”, Lobachevskii J. Math., 16 (2004), 3–15
Образцы ссылок на эту страницу:
https://www.mathnet.ru/rus/ljm81 https://www.mathnet.ru/rus/ljm/v16/p3
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