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Real closed fields and models of Peano arithmetic

Published online by Cambridge University Press:  12 March 2014

P. D'Aquino ,
J. F. Knight  and
S. Starchenko
P. D'Aquino
Affiliation:
Dipartimento di Matematica, Seconda Università di Napoli Via Vivaldi 43, Caserta 81100, Italy, E-mail: [email protected]
J. F. Knight
Affiliation:
Department of Mathematics, University of Notre Dame, 255 Hurley Hall, Notre Dame, Indiana 46556, USA, E-mail: [email protected]
S. Starchenko
Affiliation:
Department of Mathematics, University of Notre Dame, 255 Hurley Hall, Notre Dame, Indiana 46556, USA, E-mail: [email protected]
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Abstract

Shepherdson [14] showed that for a discrete ordered ring I, I is a model of I Open iff I is an integer part of a real closed ordered field. In this paper, we consider integer parts satisfyingPA. We show that if a real closed ordered fieldR has an integer part I that is a nonstandard model of PA (or even IΣ4), thenR must be recursively saturated. In particular, the real closure of I, RC (I), is recursively saturated. We also show that ifR 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.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 75 , Issue 1 , March 2010 , pp. 1 - 11
Copyright
Copyright © Association for Symbolic Logic 2012

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References

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