# Models of real numbers combined with Peano axioms

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 $\mathbb{R}$?

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: