Geodesic
Arithmetical classification of the set of all provably recursive functions
Commentationes Mathematicae Universitatis Carolinae, Tome 40 (1999) no. 4, pp. 631-634.

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The set of all indices of all functions provably recursive in any reasonable theory $T$ is shown to be recursively isomorphic to $U\times\overline{U}$, where $U$ is $\Pi_2$-complete set.
Classification : 03D20, 03D55, 03F30
Keywords: provable; recursive; complete 
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Švejdar, Vítězslav. Arithmetical classification of the set of all provably recursive functions. Commentationes Mathematicae Universitatis Carolinae, Tome 40 (1999) no. 4, pp. 631-634. http://geodesic.mathdoc.fr/item/CMUC_1999__40_4_a2/