In this paper, the authors investigate real closed fields (that is, ordered fields which are elementarily equivalent to the real numbers), looking in particular at the existence of total exponential functions on the integer part of the field. An integer part of a real closed field \(K\) is a discretely ordered subring \(Z\) such that for each \(r\in K\), there exists some \(z\in Z\) with \(z \leq r \leq z+1\). Moreover, there is a natural valuation on \(K\), in which the valuation ring \(R_v\) is the convex hull of \(\mathbb Z\) (the usual integers), and \(R_v\cap Z = \mathbb Z\).
If the field \(K\) is viewed as a structure in the language \((+,-,\cdot,0,1,<)\) of ordered rings, then \(Z\) is a substructure of \(K\), and one can ask about the theory of \(Z\). The main focus of the paper, as the title suggests, is on the case where \(Z\) is a model of Peano arithmetic (PA), although the authors do consider several other theories. In all of the considered theories, there is a formula \(E(x,y)\) which stands for \(2^x = y\). This formula satisfies many of properties one would expect from an exponential function, including “\(2^{a+b} = 2^a\cdot 2^b\)” and “for all \(x\in Z\), there exists \(y\) such that \(2^y \leq x < 2^{y+1}\).”
The paper uses these properties about exponentiation on \(Z\) to determine when a real closed field satisfies related properties about exponentiation. Specifically, let \(K\) be a non-Archimidean real closed field whose integer part \(Z\) is a model of PA (hereafter referred to as “admitting an IPA”). The main theorem concludes that:

1.

\(K\) admits left exponentiation: if \(A\) is a vector space complement of \(R_v\) in \((K,+)\) and \(B\) is a vector space complement of \(\{a\in K^{>0} : v(a)=0\}\) in \((K,\cdot)\) then there is an order-preserving group isomorphim from \(A\) onto \(B\).

2.

The value group of \(K\) is an exponential group in the additive group of the residue field: if \((G, +, -, 0, <)\) is the value group and \((\overline K, +, -, 0, <)\) is the additive group of the residue field, then the rank of \(G\) is isomorphic to the negative cone \(G^{<0}\) and the Archimidean components of \(G\) are all isomorphic (as ordered groups) to \(\overline K\).

After proving the main theorem, the paper proves a number of corollories, including constructions of real closed fields which do not admit an IPA and a proof that every model of PA is the integer part of a real closed field which admits left exponentiation. The final section of the paper includes a construction of a countable non-Archimedean real closed field which admits left exponentiation, but which is does not admit an IPA.
The paper is very well written, and the content contained in the paper is accessible for a reader with a small amount of familiarity with either real closed fields or valued fields. The only background required about models of arithmetic is an intuitive understanding of the structure of the natural numbers. Some of the proofs are referenced to other papers: most notably, part (2) of the main theorem follows from part (1) and two results from [S. Kuhlmann, Ordered exponential fields. Providence, RI: American Mathematical Society (AMS) (2000; Zbl 0989.12003)]. The final construction uses a small amount of computability theory, but the main results can be understood without a strong background in computability.