Natural exponentially transcendental numbers and the decidability of the real field expanded by a natural irrational power function

Jones and Servi (2011) proved that if a real number $\alpha$ is generic — equivalently, exponentially transcendental, i.e. not first-order definable without parameters in the real exponential field $\mathbb{R}_{\exp}$ — then the theory of the expansion $\mathbb{R}^\alpha=\langle\mathbb{R};+,-,\cdot,0,1,<,\alpha,x\mapsto x^\alpha\rangle$ is decidable relative to an oracle for the rational cut of $\alpha$, and they constructed a computable generic $\alpha$ by diagonalization, yielding the first example (found independently by D. J. Miller) of a decidable proper o-minimal expansion of the real field by a transcendental analytic function. The constructed exponent is an artificial diagonalization number, which the authors themselves describe as 'far from being explicit'. The problem left open by that work is whether any classically defined ('natural') constant — $\pi$ above all — is exponentially transcendental, and whether the corresponding power-function expansion, in particular $\mathbb{R}^{\pi}$, has decidable complete theory. Since $e=\exp(1)$ is trivially 0-definable in $\mathbb{R}_{\exp}$ and hence not generic, the natural-constant question concerns $\pi$ and similar constants. The definability of $\pi$ in $\mathbb{R}_{\exp}$ and the decidability of $\operatorname{Th}(\mathbb{R}^{\pi})$ remain open.

Background

The real exponential field and Tarski's problem. The real exponential field is the structure $\mathbb{R}_{\exp}=(\mathbb{R};+,-,\cdot,0,1,<,\exp)$: the real ordered field expanded by the real exponential function. Tarski proved that the first-order theory of the real ordered field alone is decidable (an algorithm determines exactly which sentences are true in $\mathbb{R}$), and asked whether decidability survives the addition of exponentiation. Wilkie (1996) proved that $\mathbb{R}_{\exp}$ is model complete (every first-order formula is equivalent, in all models of its theory, to an existential formula) and o-minimal — 'o-minimal' means every definable subset of $\mathbb{R}$ is a finite union of intervals and points, so all definable sets have tame, semialgebraic-style geometry. The decidability of the complete theory $\operatorname{Th}(\mathbb{R}_{\exp})$ is still open. Macintyre and Wilkie (1996) proved decidability assuming Schanuel's conjecture from transcendence theory: for $\mathbb{Q}$-linearly independent reals $x_1,\dots,x_n$, the $2n$ numbers $x_1,\dots,x_n,e^{x_1},\dots,e^{x_n}$ have transcendence degree at least $n$ over $\mathbb{Q}$. Schanuel's conjecture is considered out of reach — even the irrationality of $e^e$ is unknown.

Power functions and generic exponents. Seeking some decidable expansion of $\mathbb{R}$ by a transcendental analytic function, Jones and Servi (2011) studied, for a real number $\alpha$, the power function $x^\alpha$ (defined as $\exp(\alpha\log x)$ for $x>0$ and $0$ elsewhere) and the expansion $\mathbb{R}^\alpha=\langle\mathbb{R};+,-,\cdot,0,1,<,\alpha,x^\alpha\rangle$, which adds both a constant for $\alpha$ and the power function; foundational results of C. Miller (1994) on expansions of the real field by power functions supply the model-completeness and effective ('rational type') axiomatics for the restricted power function on which their argument builds. Call $\alpha$ exponentially algebraic if it occurs as the first coordinate of a regular zero — a common zero at which the Jacobian determinant is nonzero — of a system of exponential polynomials with integer coefficients in the variables $x_1,\dots,x_n,e^{x_1},\dots,e^{x_n}$; otherwise $\alpha$ is exponentially transcendental. A theorem of Jones and Wilkie (2008) identifies exponentially transcendental numbers with generic ones, i.e. those not definable without parameters (0-definable) in $\mathbb{R}_{\exp}$. Bays, Kirby and Wilkie (2010) proved that all but countably many reals are exponentially transcendental, and that such exponents satisfy a Schanuel-type inequality for powers ($\operatorname{trdeg}_{\mathbb{Q}(\alpha)}\mathbb{Q}(\alpha)(a_1,\dots,a_n,a_1^\alpha,\dots,a_n^\alpha)\ge n$ for multiplicatively independent positive reals $a_i$ (no relation $a_1^{k_1}\cdots a_n^{k_n}=1$ with integers $k_i$ not all zero)), which is the ingredient that lets the Macintyre–Wilkie argument run for $\mathbb{R}^\alpha$. The main theorem of Jones–Servi then states: if $\alpha$ is generic, $\operatorname{Th}(\mathbb{R}^\alpha)$ is decidable relative to an oracle deciding the rational (Dedekind) cut of $\alpha$ in $\mathbb{Q}$, i.e. deciding, for each rational $q$, whether $q<\alpha$. Finally, by diagonalizing against an effective enumeration of integer exponential-polynomial systems that have regular zeros, they produced a computable exponentially transcendental $\alpha$; a computable $\alpha$ has a decidable cut, so the theory is then decidable outright. This gave (independently of D. J. Miller's construction via generic families of functions from quasianalytic classes, i.e. classes of smooth functions determined locally by their Taylor expansions) the first examples of a decidable proper o-minimal expansion of $\mathbb{R}$ by an analytic function not definable in the pure field.

What is missing. The diagonalization exponent is by design an artificial number; the paper says it 'is rather far from being explicit'. Among classical constants the situation is rigid at one end and dark at the other: $e=\exp(1)$, $\log 2$, $e^e$, and everything generated from $\mathbb{Q}$ by field operations, $\exp$ and $\log$ are 0-definable in $\mathbb{R}_{\exp}$, hence exponentially algebraic, hence not generic. For $\pi$ — and every other classical constant not so generated (Euler–Mascheroni constant $\gamma$, Apéry's constant $\zeta(3)$, Catalan's constant, $\Gamma(1/4)$, …) — it is unknown whether it is 0-definable in $\mathbb{R}_{\exp}$, and unknown whether $\operatorname{Th}(\mathbb{R}^{\pi})$ is decidable. The oracle in the Jones–Servi theorem is inherent rather than an artifact: $\mathbb{R}^\alpha$ contains a constant for $\alpha$, so its theory computes the cut of $\alpha$, and for non-computable $\alpha$ the theory is undecidable in the usual sense; for classical constants such as $\pi$ the cut is decidable, so the only obstruction to applying the theorem is genericity. Recent work on the surrounding landscape — Berarducci and Gallinaro's 2026 re-derivation of the Macintyre–Wilkie conditional decidability, and tameness results for expansions by power functions and multiplicative subgroups (Hieronymi; Zhang) — leaves both questions untouched.

Problem Statement

Let $\mathbb{R}^{\pi}=\langle\mathbb{R};+,-,\cdot,0,1,<,\pi,x\mapsto x^{\pi}\rangle$, where $x^{\pi}=\exp(\pi\log x)$ for $x>0$ and $x^{\pi}=0$ elsewhere. The problem arising from the results and the acknowledged limitations of Jones–Servi (2011) has two linked parts serving one objective, replacing the diagonalization-constructed generic exponent by a naturally occurring constant:

1. Natural generic constants. Does there exist a classically defined ('natural') real number $\alpha$ — for instance $\pi$, the Euler–Mascheroni constant $\gamma$, or Apéry's constant $\zeta(3)$ — that is exponentially transcendental, i.e. not the first coordinate of any regular zero (a common zero at which the Jacobian determinant is nonzero) of any system of exponential polynomials with integer coefficients in variables $x_1,\dots,x_n,e^{x_1},\dots,e^{x_n}$? By the Jones–Wilkie equivalence this is the same as asking whether such a constant is not 0-definable in the real exponential field $\mathbb{R}_{\exp}$. Part 1 is thus the general existence question, resolved by exhibiting any one natural exponentially transcendental constant, $\pi$ being the flagship target. The flagship instance: is $\pi$ exponentially transcendental, i.e. is $\pi$ not first-order definable without parameters in $\mathbb{R}_{\exp}$? (A negative answer for the flagship instance, consisting of an explicit parameter-free first-order definition of $\pi$ in $\mathbb{R}_{\exp}$ together with a proof of its uniqueness, settles it in the opposite direction.)

2. Decidability of the natural expansion. Is the complete first-order theory $\operatorname{Th}(\mathbb{R}^{\pi})$ decidable, in the usual sense of a recursive set of true sentences? By the main theorem of Jones–Servi together with the computability of $\pi$ (which trivializes the oracle for the rational cut of $\pi$), a positive answer to part 1 for $\alpha=\pi$ settles part 2 as well; part 2 may also be settled independently of part 1.

The common scientific objective of the two parts is to obtain a naturally specified instance of a decidable proper o-minimal expansion of the real field by a transcendental analytic function — the constructive aim stated by Jones and Servi, which their diagonalized exponent by their own admission does not fulfill, leaving the artificial exponent to be replaced by a natural one — or to settle definitively, in either direction, the two reconstructed yes/no questions above for a classical constant. (Here 'natural' means a constant of classical analysis or number theory — $\pi$, $\gamma$, $\zeta(3)$, Catalan's constant, $\Gamma(1/4)$ and the like — defined without reference to the halting behaviour of programs, to an encoding of an undecidable set, or to a diagonalization; this is the class that part 1 ranges over. Constants generated from $\mathbb{Q}$ by field operations, $\exp$ and $\log$, such as $e$, $\log 2$ or $e^e$, are already settled as exponentially algebraic and do not qualify.)

References

1. Gareth Jones and Tamara Servi, 'On the decidability of the real field with a generic power function', Journal of Symbolic Logic 76(4) (2011), pp. 1418–1428. DOI: 10.2178/jsl/1318338857. https://doi.org/10.2178/jsl/1318338857 (preprint: MIMS EPrint 2011.55, http://eprints.maths.manchester.ac.uk/1643/)
2. Martin Bays, Jonathan Kirby, and A. J. Wilkie, 'A Schanuel property for exponentially transcendental powers', Bulletin of the London Mathematical Society 42(5) (2010), pp. 917–922. DOI: 10.1112/blms/bdq054. https://doi.org/10.1112/blms/bdq054 (preprint: arXiv:0810.4457, https://arxiv.org/abs/0810.4457)
3. G. O. Jones and A. J. Wilkie, 'Locally polynomially bounded structures', Bulletin of the London Mathematical Society 40(2) (2008), pp. 239–248. DOI: 10.1112/blms/bdn004. https://doi.org/10.1112/blms/bdn004
4. Angus Macintyre and A. J. Wilkie, 'On the decidability of the real exponential field', in P. Odifreddi (ed.), Kreiseliana: About and Around Georg Kreisel, A. K. Peters, Wellesley, MA, 1996, pp. 441–467. ISBN 978-1-56881-061-4. zbMATH review: https://zbmath.org/?q=an%3A0896.03012
5. A. J. Wilkie, 'Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function', Journal of the American Mathematical Society 9(4) (1996), pp. 1051–1094. DOI: 10.1090/S0894-0347-96-00216-0. https://doi.org/10.1090/S0894-0347-96-00216-0
6. Chris Miller, 'Expansions of the real field with power functions', Annals of Pure and Applied Logic 68(1) (1994), pp. 79–94. DOI: 10.1016/0168-0072(94)90048-5. https://doi.org/10.1016/0168-0072(94)90048-5
7. Daniel J. Miller, 'Constructing o-minimal structures with decidable theories using generic families of functions from quasianalytic classes', preprint (2010). arXiv:1008.2575. https://arxiv.org/abs/1008.2575
8. Philipp Hieronymi, 'The real field with an irrational power function and a dense multiplicative subgroup', Journal of the London Mathematical Society (2) 83(1) (2011), pp. 153–167. DOI: 10.1112/jlms/jdq058. https://doi.org/10.1112/jlms/jdq058
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10. Joseph Harrison, 'Additive relations in irrational powers', preprint (2025). arXiv:2512.04081. https://arxiv.org/abs/2512.04081
11. Alessandro Berarducci and Francesco Gallinaro, 'On the elementary theory of the real exponential field', preprint (2026). arXiv:2603.08365. https://arxiv.org/abs/2603.08365
12. Yilong Zhang, 'Raising to powers on the unit circle', preprint (2026). arXiv:2608.08655. https://arxiv.org/abs/2608.08655
13. Will Johnson, 'Weakly o-minimal fields have the exchange property but not generic differentiability', preprint (2026). arXiv:2606.08527. https://arxiv.org/abs/2606.08527
14. Vahagn Aslanyan, Sebastian Eterović, and Jonathan Kirby, 'A closure operator respecting the modular j-function', Israel Journal of Mathematics 253(1) (2022), pp. 321–357. DOI: 10.1007/s11856-022-2362-y. https://doi.org/10.1007/s11856-022-2362-y (preprint: arXiv:2010.00102, https://arxiv.org/abs/2010.00102)