Geodesic
On~arithmetic complexity of the~predicate logics of complete constructive arithmetic theories
Fundamentalʹnaâ i prikladnaâ matematika, Tome 5 (1999) no. 1, pp. 221-255

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It is proved in this paper that the predicate logic of each complete constructive arithmetic theory $T$ having the existence property is $\Pi_1^T$-complete. In this connection the techniques of uniform partial truth definition for intuitionistic arithmetic theories is used. The main theorem is applied to the characterization of the predicate logic corresponding to certain variant of the notion of realizable predicate formula. Namely it is shown that the set of undisprovable predicate formulas is recursively isomorphic to the complement of the set $\emptyset^{(\omega +1)}$. 
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     author = {V. E. Plisko},
     title = {On~arithmetic complexity of the~predicate logics of complete constructive arithmetic theories},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
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V. E. Plisko. On~arithmetic complexity of the~predicate logics of complete constructive arithmetic theories. Fundamentalʹnaâ i prikladnaâ matematika, Tome 5 (1999) no. 1, pp. 221-255. http://geodesic.mathdoc.fr/item/FPM_1999_5_1_a12/