Geodesic
Generalized realizability for extensions of arithmetic language
Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 4 (2019), pp. 50-54

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Let $L$ be an extension of the language of arithmetic, $V$ be a class of number-theoretical functions. A notion of the $V$-realizability for $L$-formulas is defined in such a way that indexes of functions in $V$ are used for interpreting the implication and the universal quantifier. It is proved that the semantics for $L$ based on the $V$-realizability coincides with the classic semantics iff $V$ contains all $L$-definable functions. 
@article{VMUMM_2019_4_a7,
     author = {A. Yu. Konovalov},
     title = {Generalized realizability for extensions of arithmetic language},
     journal = {Vestnik Moskovskogo universiteta. Matematika, mehanika},
     pages = {50--54},
     publisher = {mathdoc},
     number = {4},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VMUMM_2019_4_a7/}
}
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A. Yu. Konovalov. Generalized realizability for extensions of arithmetic language. Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 4 (2019), pp. 50-54. http://geodesic.mathdoc.fr/item/VMUMM_2019_4_a7/