✦ LIBER ✦

Nonstandard models that are definable in models of Peano Arithmetic

✍ Scribed by Kazuma Ikeda; Akito Tsuboi

Publisher: John Wiley and Sons
Year: 2007
Tongue: English
Weight: 183 KB
Volume: 53
Category: Article
ISSN: 0044-3050
DOI: 10.1002/malq.200610020

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✦ Synopsis

Abstract

In this paper, we investigate definable models of Peano Arithmetic PA in a model of PA. For any definable model N without parameters in a model M, we show that N is isomorphic to M if M is elementary extension of the standard model and N is elementarily equivalent to M. On the other hand, we show that there is a model M and a definable model N with parameters in M such that N is elementarily equivalent to M but N is not isomorphic to M. We also show that there is a model M and a definable model N with parameters in M such that N is elementarily equivalent to M, and N is isomorphic to M, but N is not definably isomorphic to M. And also, we give a generalization of Tennenbaum's theorem. At the end, we give a new method to construct a definable model by a refinement of Kotlarski's method. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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