Analogues of Shepherdson’s Theorem for a language with exponentiation

Abstract In 1964, Shepherdson (1964, Bull. Pol. Acad. Sci., 12) proved that a discretely ordered semiring $\mathcal{M}^{+}$ satisfies $\textsf{IOpen}$ (quantifier-free induction) iff the corresponding ring $\mathcal{M}$ is an integer part of a model of the theory of real closed fields (RCF). In this paper, we consider open induction schema in the language of arithmetic expanded by exponentiation or by the power function and try to find similar criteria for models of these theories. For several recursively axiomatized extensions $T$ of the theory of RCF, we obtain analogues of Shepherdson’s Theorem in the following sense: If an exponential field $\mathcal R$ is a model of $T$ and a discretely ordered ring (DOR) $\mathcal M$ is an exponential integer part of $\mathcal R$, then $\mathcal M^{+}$ is a model of open induction in the expanded language. The proof of the opposite implication—that for any model $\mathcal M$ of open induction in the expanded language there exists an exponential field $\mathcal R \vDash T$ such that $\mathcal M$ is an exponential integer part of $\mathcal R$—remains, in general, an open question. However, we isolate a natural sufficient condition, related to the well-known Bernoulli inequality, under which this result holds. We define a finite extension $T$ of the usual open induction so that, for any DOR $\mathcal M$, the semiring $\mathcal M^{+}$ satisfies $T$ iff there is an exponential RCF $\mathcal R$ with the inequality $\exp (x) \geqslant 1 + x$ such that $\mathcal M$ is an exponential integer part of $\mathcal R$. Using these results, we obtain some concrete independence results for these theories.