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Suppose you take the axioms for a Dedekind-complete ordered field and weaken the Dedekind-completeness axiom to the corresponding weaker first-order axiom schema (e.g. replace the left and right sets of the cuts with first-order formulae in the language of ordered fields). Then, I believe, the resulting theory is that of real-closed fields, which are the models.

But now, what happens if we attempt to add to this theory an additional predicate for the natural numbers, and add Peano's axioms (defining $0$ (the field additive identity) to be a natural number and successor by $S(n) = n + 1$ where $1$ is the field's multiplicative identity, and adding the Peano axioms including the first-order induction schema)? What are the models of this combined theory? Does any real-closed field work as a model provided a suitable definition of natural number is made upon it? If not, then what would the models be like, especially the ones larger than $\mathbb{R}$?

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It is not the case that the class of all real-closed fields works in this case. There is a classical theorem due to Shepherdson that in general, a model of $M$ in the language of Peano arithmetic is the integer part of a real-closed field if and only if $M$ satisfies the induction scheme for quantifier-free formulas (also called "open induction" or "IOpen"), which is much weaker than the induction scheme of Peano arithmetic. The paper you are interested in is:

P. D'Aquino, J. F. Knight, and S. Starchenko, "Real closed fields and models of Peano arithmetic", Journal of Symbolic Logic, v. 75 n. 1, 2010, pp. 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.

Basically, to get the integer part of a real-closed field to be a model of PA, we need to assume a stronger property of the field, which is known in model theory as "recursive saturation".

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