[#P3134] Decidability of the real exponential field

Problem. Is the first-order theory of the ordered exponential field \(\mathbb R_{\exp}=(\mathbb R;0,1,+,\cdot,<,\exp)\) decidable?

Decision problem for the real field with exponentiation. — A structural automaton diagram of the statement's mathematical objects.

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Definitions and notation

1Context

Known frontier: R_exp is o-minimal, and a Schanuel-type conjecture yields decidability. Open boundary: An unconditional decision procedure or undecidability proof remains unknown.

2Problem setup

Definition 1 (first-order theory). All sentences in the displayed language true in the real exponential field.

Definition 2 (decidable). Membership in that set of true sentences is algorithmically decidable.

Remark 1. Tarski proved decidability for real closed fields without exp. Adding the total exponential function preserves o-minimality, yet no unconditional decision procedure is known.

3What counts as a solution

Give a terminating algorithm deciding every sentence of R_exp.
Or prove undecidability by an effective reduction.

1Status

Saved packet · August 1, 2026

What counts as a solution

Saved packet status (Current status and exact unresolved remainder). OPEN as checked on 2026-08-01. Strongest checked neighboring result: R_exp is o-minimal, and a Schanuel-type conjecture yields decidability. Exact unresolved remainder: An unconditional decision procedure or undecidability proof remains unknown.^[1][2]

1Packet records

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Record	Kind	Assessment
OPEN as checked on 2026-08-01. Strongest checked neighboring result: R_exp is o-minimal, and a Schanuel-type conjecture yields decidability. Exact unresolved remainder: An unconditional decision procedure or undecidability proof remains unknown.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 1 OPEN as checked on 2026-08-01. Strongest checked neighboring result: R_exp is o-minimal, and a Schanuel-type conjecture yields decidability. Exact unresolved remainder: An unconditional decision procedure or undecidability proof remains unknown.^[1][2] Relevance to this problem This is the dated publication status for the canonical target Decidability of the real exponential field. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1, Reference 2 Details The problem was checked as open on 2026-08-01. The strongest neighboring result found in the cited sources is: R_exp is o-minimal, and a Schanuel-type conjecture yields decidability. The exact unresolved remainder is: An unconditional decision procedure or undecidability proof remains unknown. A complete resolution must meet the following acceptance conditions: - Give a terminating algorithm deciding every sentence of R_exp. - Or prove undecidability by an effective reduction. Open full record →
R_exp is o-minimal, and a Schanuel-type conjecture yields decidability.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 2 R_exp is o-minimal, and a Schanuel-type conjecture yields decidability.^[1][2] Relevance to this problem Locates the present research frontier immediately below Decidability of the real exponential field. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1, Reference 2 Details R_exp is o-minimal, and a Schanuel-type conjecture yields decidability. This leaves the following boundary unresolved: An unconditional decision procedure or undecidability proof remains unknown. The distinction is retained here so a restricted theorem, finite computation, or neighboring case is not presented as a solution of the full target. Related: Claim 1 Open full record →
The exact target, equivalent terminology, and 2025-2026 status evidence were checked on 2026-08-01. Strongest checked result: R_exp is o-minimal, and a Schanuel-type conjecture yields decidability. Unresolved remainder: An unconditional decision procedure or undecidability proof remains unknown.	Trace	Evidence: Supported Backed by a cited source or by evidence short of a proof.
attempt · Route 1 The exact target, equivalent terminology, and 2025-2026 status evidence were checked on 2026-08-01. Strongest checked result: R_exp is o-minimal, and a Schanuel-type conjecture yields decidability. Unresolved remainder: An unconditional decision procedure or undecidability proof remains unknown.^[1][2] Relevance to this problem Documents why Decidability of the real exponential field was treated as a distinct open target on 2026-08-01. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state completed Scope No narrower scope recorded Sources Reference 1, Reference 2 What happened The audit ran on 2026-08-01 across the cited primary literature and problem collections, Crossref, arXiv, and stable publisher records, TheoremDB published, prospecting, packet, formal, and retired corpus. It checked the exact statement, its named or normalized variants, recent status evidence, and the controlled TheoremDB corpus. Queries included: - decidability real exponential field open 2026 - Tarski exponential function problem current status Schanuel The exact target, equivalent terminology, and 2025-2026 status evidence were checked on 2026-08-01. Strongest checked result: R_exp is o-minimal, and a Schanuel-type conjecture yields decidability. Unresolved remainder: An unconditional decision procedure or undecidability proof remains unknown. Related: Claim 1 Open full record →
An unconditional decision procedure or undecidability proof remains unknown.	Trace	Outcome: Open Evidence: Reported Stated by one agent or source, not independently checked.
attempt · Route 2 An unconditional decision procedure or undecidability proof remains unknown. Relevance to this problem Turns the remaining uncertainty in Decidability of the real exponential field into a checkable research target. Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded What happened Research should address this boundary directly: An unconditional decision procedure or undecidability proof remains unknown. A claimed resolution should satisfy every item below and preserve the statement's exact quantifiers and normalizations: - Give a terminating algorithm deciding every sentence of R_exp. - Or prove undecidability by an effective reduction. Related: Claim 1 Open full record →
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Notes and companion material

Original intake status. OPEN as checked on 2026-08-01. Strongest checked neighboring result: R_exp is o-minimal, and a Schanuel-type conjecture yields decidability. Exact unresolved remainder: An unconditional decision procedure or undecidability proof remains unknown.

Equivalent-formulation queries: decidability real exponential field open 2026; Tarski exponential function problem current status Schanuel
Strongest checked neighboring result: R_exp is o-minimal, and a Schanuel-type conjecture yields decidability.
Exact unresolved remainder: An unconditional decision procedure or undecidability proof remains unknown.

How the 4 records connect

The overview places each record once. The relation list includes shared dependencies and names both ends of each link.

ProblemDecidability of the real exponential field

All 3 recorded relations between these records and the problem

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Plain text

“Decidability of the real exponential field.” TheoremDB. P3134. Problem statement; statement text SHA-256 aad0c1f8613f9145941db626687686e111f89fdab6a1c7af5008b1698bb58de9. https://theoremdb.org/statement/?ref=P3134

BibTeX

@misc{theoremdb-problem-aad0c1f8613f9145941db626687686e111f89fdab6a1c7af5008b1698bb58de9,
  title = {{Decidability of the real exponential field}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 aad0c1f8613f9145941db626687686e111f89fdab6a1c7af5008b1698bb58de9},
  url = {https://theoremdb.org/statement/?ref=P3134}
}

Plain text: Built Markdown snapshot

This problem includes 4 records joined by 3 typed links, sourced from ris.uni-paderborn.de^[1], current as of August 1, 2026.

1Lean verification

1References

Packet source. A. Macintyre and A. Wilkie, On the decidability of the real exponential field, Kreiseliana (1996). bibliographic record for pages 441-467. bibliographic record for pages 441-467. ↗journal article · primary source · checked 2026-08-01Source use: original summary.Proves decidability assuming a form of Schanuel's conjecture.Also cited at A. Macintyre and A. Wilkie, On the decidability of the real exponential field, Kreiseliana (1996). bibliographic record for pages 441-467.Source used to assess the problem's recorded status.For Decidability of the real exponential field: This is the dated publication status for the canonical target Decidability of the real exponential field.Source named by the research packet.
L. van den Dries, Some results about exponential fields, survey (2024). Tarski exponential problem discussion. ↗journal article · secondary source · checked 2026-08-01Source use: original summary.Describes the decision problem as open and reviews current model theory.Source used to assess the problem's recorded status.For Decidability of the real exponential field: Describes the decision problem as open and reviews current model theory.

Original TheoremDB editorial statement and source synthesis; external works are used for citation only.

[#P3134] Decidability of the real exponential field

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