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Generalized realizability for extensions of arithmetic language

A. Yu. Konovalov

Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: 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.

Key words: constructive semantics, realizability, generalized realizability, formal arithmetic.

UDC: 510.25+510.64

Received: 04.07.2018

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