Tarski's exponential function problem

About 78 years old · traced to 1948

Let $L_{\text{or}}=(+,-,\cdot,<,0,1)$ be the language of ordered rings, and let $L_{\exp}=L_{\text{or}}\cup\{\exp\}$ , where $\exp$ is a unary function symbol. Let $\mathbb{R}_{\exp}$ denote the $L_{\exp}$ -structure with underlying set $\mathbb{R}$ in which the symbols of $L_{\text{or}}$ receive their usual interpretations in the ordered real field and $\exp$ is interpreted as the real exponential function $x\mapsto e^{x}$ . Let

\operatorname{Th}(\mathbb{R}_{\exp})=\{\varphi : \varphi \text{ is an } L_{\exp}\text{-sentence and } \mathbb{R}_{\exp}\models\varphi\}.

Then $\operatorname{Th}(\mathbb{R}_{\exp})$ is decidable: there is an effective procedure which, given any $L_{\exp}$ -sentence $\varphi$ , determines whether

\mathbb{R}_{\exp}\models\varphi.

References

Primary source

Wikipedia

Additional references

Wikipedia, Tarski's exponential function problem, the article this problem comes from.

logic unsolved-problem

Posted 2 months ago

Progress summary

Refreshed 1 month ago

Open

The decision problem is still open, with only conditional results and no verified proof or counterexample.

Tarski’s problem asks whether the first-order theory of the real numbers with addition, multiplication, order, and exponential function is decidable. Tarski had already established decidability for the real field without exponential function.

Known results

Wilkie, 1996: the real exponential field is model-complete.
Macintyre and Wilkie, 1996: decidability follows from the real version of Schanuel’s conjecture; a weaker equivalent formulation is also known.
van den Dries, 1998: the real exponential field does not admit quantifier elimination.
Berarducci and Servi, 2006: conditional decidability follows from the Transfer Conjecture.

November 2024 update

A preprint on finite models for exponential algebra explicitly states that Tarski’s exponential-function problem remains open and does not provide a direct solution. No verified proof, counterexample, correction, or AI attribution was found.

Current status (as of August 2026): Decidability of $\operatorname{Th}(\mathbb{R}_{\exp})$ remains unsettled; the principal result is conditional on Schanuel-type conjectures.

Sources

No solutions have been posted yet.