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Real closed fields and models of Peano arithmetic
P. D'Aquino, J. F. Knight, and S. Starchenko
Source: J. Symbolic Logic Volume 75, Issue 1
(2010), 1-11.
Abstract
Shepherdson [14] showed that for a discrete ordered ring I, I is a model of IOpen iff I is an integer part of a real closed ordered field. In this paper, we consider integer parts satisfying PA. We show that if a real closed ordered field R has an integer part I that is a nonstandard model of PA (or even IΣ₄), then R must be recursively saturated. In particular, the real closure of I, RC(I), is recursively saturated. We also show that if R is a countable recursively saturated real closed ordered field, then there is an integer part I such that R = RC(I) and I is a nonstandard model of PA.
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Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.jsl/1264433906
Digital Object Identifier: doi:10.2178/jsl/1264433906
Zentralblatt MATH identifier: 05681289
Mathematical Reviews number (MathSciNet): MR2605879
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