Elementary Equivalences for Ordered Exponential Fields

Definitions and mathematical statement

Let $\mathcal R=(R;0,1,+,-,\cdot,<,\exp)$ be an expansion of an ordered field by a unary function $\exp:R\to R$. Equip $R$ with its order topology. Assume that $\exp$ is everywhere continuous and differentiable, with \[ \exp(0)=1,\qquad \exp'(x)=\exp(x)\quad\text{for every }x\in R. \] Here differentiability is understood over the ordered field: for every $x\in R$ and $\varepsilon>0$, there exists $\delta>0$ such that \[ 0<|h|<\delta\quad\Longrightarrow\quad \left|\frac{\exp(x+h)-\exp(x)}{h}-\exp(x)\right|<\varepsilon. \] No Archimedean assumption is imposed on $R$.

All definability below is in the displayed language and allows parameters from $R$. The structure $\mathcal R$ is o-minimal if every definable subset of $R$ is a finite union of singletons and open intervals, allowing interval endpoints in $R\cup\{-\infty,+\infty\}$. It is definably complete if every nonempty definable subset $A\subseteq R$ that is bounded above has a least upper bound in $R$.

Let \[ \mathbb R_{\exp}=(\mathbb R;0,1,+,-,\cdot,<,x\mapsto e^x) \] be the usual real exponential field. Elementary equivalence $\mathcal R\equiv\mathbb R_{\exp}$ means that these structures satisfy exactly the same first-order sentences in the displayed language.

The question is whether, for every $\mathcal R$ satisfying the hypotheses above, the following conditions are equivalent:

1. $\mathcal R\equiv\mathbb R_{\exp}$.
2. $\mathcal R$ is o-minimal.
3. $\mathcal R$ is definably complete.

The function hypotheses are taken literally from the supplied clean statement. Additional axioms such as $\exp(x+y)=\exp(x)\exp(y)$, positivity, or surjectivity onto the positive elements are not separately assumed in that statement; whether the report's surrounding context imposes additional hypotheses requires review. The supplied literature assessment describes the unconditional equivalence as open and refers to a conditional implication using Schanuel-type reasoning, but does not specify the exact conditional hypothesis.

Short English statement

For every ordered field equipped with the continuous differentiable function specified above, ask whether elementary equivalence to the usual real exponential field, o-minimality, and definable completeness are equivalent. Definability includes the function $\exp$ and parameters. The supplied record labels the problem open; its literature assessment and source-verification claims are supplied curation, not independent verification by this draft.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30006584$, title Elementary Equivalences for Ordered Exponential Fields, exact source problem code catalogue entry, in the collection Oberwolfach Reports Open Problems. The record assigns year $2026$ and supplies no individual proposer.
• Model Theory and Arithmetic (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/3, pp. 157--216: https://doi.org/10.4171/owr/2026/3. This is also the supplied literature source labelled "Oberwolfach Report 3/2026." No pinpoint page for this problem is supplied.
• Supplied literature source: On the elementary theory of the real exponential field, https://arxiv.org/abs/2603.08365. The supplied record does not give its authors or a specific theorem reference.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open, reports that two implications are established, and describes the remaining implication from definable completeness to the complete theory of $\mathbb R_{\exp}$ as conditional on Schanuel-type reasoning. This is supplied dated curation, not an independently verified literature assessment; the source itself requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports that the official Oberwolfach report was checked directly, with OCR, notation, and source-dependent wording repaired while preserving the target. These are supplied source-verification claims; no source consultation is claimed by this draft.