Geodesic
The nonarithmeticity of the class of realizable predicate formulas
Izvestiya. Mathematics , Tome 11 (1977) no. 3, pp. 453-471

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It is proved that the set of Gödel numbers of realizable predicate formulas is not definable in the language of formal arithmetic. Bibliography: 13 titles. 
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     author = {V. E. Plisko},
     title = {The nonarithmeticity of the class of realizable predicate formulas},
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V. E. Plisko. The nonarithmeticity of the class of realizable predicate formulas. Izvestiya. Mathematics , Tome 11 (1977) no. 3, pp. 453-471. http://geodesic.mathdoc.fr/item/IM2_1977_11_3_a0/