Nonstandard models that are definable in models of Peano Arithmetic |
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Authors: | Kazuma Ikeda Akito Tsuboi |
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Affiliation: | 1. Faculty of Humanities, Seitoku University, 550 Iwase, Matsudo-shi, Chiba, 271-8555, Japan;2. Institute of Mathematics, University of Tsukuba, Tsukuba-shi, Ibaraki, 305-8571, Japan |
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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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Keywords: | Peano Arithmetic definable model Tennenbaum's theorem |
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