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Real closed fields and models of Peano arithmetic. (English) Zbl 1186.03061

J. Symb. Log. 75, No. 1, 1-11 (2010); corrigendum ibid. 77, No. 2, 726 (2012).
An integer part of an ordered field \(R\) is a discretely ordered subring \(I\) such that 1 is the least positive element, and for each \(x\in R\) there is some \(i\in I\) such that \(i\leq x <i+1\). By a classical result of J. C. Shepherdson [Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 12, 79–86 (1964; Zbl 0132.24701)] a discretely ordered ring is an integer part of a real closed field if and only if it is a model of Open Induction. In this paper the authors ask for conditions under which a real closed field has an integer part that is a model of PA. There are two main results: (1) If \(R\) is a real closed field with an integer part that is a model of PA (or even \(I\Sigma_4\)), then \(R\) is recursively saturated. If, in addition, \(R\) is countable, and \(I\models\) PA is its integer part, then \(R\) is isomorphic to the real closure of \(I\). (2) If \(R\) is a recursively saturated countable real closed field, then \(R\) has an integer part satisfying PA.

MSC:

03C62 Models of arithmetic and set theory
03H15 Nonstandard models of arithmetic

Citations:

Zbl 0132.24701
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References:

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