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The $V$-realizability for $L$-formulas coincides with the classical semantics if $V$ contains all $L$-definable functions

A. Yu. Konovalov

Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: Let $L$ be an extension of the language of arithmetic, $V$ 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 classical semantics if $V$ contains all $L$-definable functions.

Keywords: constructive semantics, realizability, generalized realizability, formal arithmetic.