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.


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


Zbl 0132.24701
Full Text: DOI


[1] DOI: 10.1051/ita:2007048 · Zbl 1144.03027
[2] Logic, algebra and arithmetic 26 pp 42– (2006)
[3] An introduction to recursively saturated and resplendent models 41 pp 531– (1976) · Zbl 0343.02032
[4] Bulletin de l’Academic Polonaise des Sciences 12 pp 79– (1964)
[5] Recursive function theory pp 117– (1962)
[6] Logic Colloquium ’80 (Prague, 1980) pp 57– (1982)
[7] Lectures in abstract algebra. Volume III (1975)
[8] Annals of Pure and Applied Logic 12 pp 151– (1977)
[9] DOI: 10.1090/S0002-9947-1986-0833697-X
[10] Arithmetic, proof theory, and computational complexity (1993) · Zbl 0777.00008
[11] Every real closed field has an integer part 58 pp 641– (1993)
[12] Proceedings of the American Mathematical Society 68 pp 331– (1978)
[13] DOI: 10.1090/S0002-9947-1986-0833698-1
[14] DOI: 10.1305/ndjfl/1093870818 · Zbl 0487.03018
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.