## 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

Zbl 0132.24701
Full Text:

### References:

 [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
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