In 1948 Tarski posed the Following question: let $\mathbb{R}{\text{exp}}$ be the expansion of the ordered ring of the reals with the exponential function. Is the complete theory $T$ decidable?}}$ of $\mathbb{R}_{\text{exp}
In 1984, attempting to tackle this question geometrically led van den Dries to introduce the notion now known as o-minimality, with Wilkie later proving that $\mathbb{R}{\text{exp}}$ is model complete, and thus, by earlier results of Khovanskii, that $\mathbb{R}$ is decidable, provided that the real Schanuel's conjecture is true. Thus, the problem has been reduced to a far-reaching conjecture in transcendental number theory which seems unlikely to be solved any time soon.}}$ is indeed o-minimal. Finally Macintyre and Wilkie proved in 1996 that $T_{\text{exp}
However, in the late forties, Kleene and Mostowski independently introduced a hierarchy of undecidabilities now known as the arithmetical hierarchy. We will place $T_{\text{exp}}$ in the arithmetical hierarchy providing a low upper bound by using the tools of effective o-minimality. We will contrast the case of $\texy{exp}$ with that of other real analytic functions and real computable functions. The latter case provably exhibits a great variety of possible computational complexities.
(This talk is based on a paper in preparation by the speaker)