Abstract
AbstractAn arithmetical interpretation of dynamic propositional logic (DPL) is a mapping ƒ satisfying the following: (1) ƒ associates with each formula A of DPL a sentence ƒ(A) of Peano arithmetic (PA) and with each program α a formula ƒ(α) of PA with one free variable describing formally a supertheory of PA; (2) ƒ commutes with logical connectives; (3) ƒ([α]A) is the sentence saying that ƒ(A) is provable in the theory ƒ(α); (4) for each axiom A of DPL, ƒ(A) is provable in PA (and consequently, for each A provable in DPL, ƒ(A) is provable in PA). The arithmetical completeness theorem is proved saying that a formula A of DPL is provable in DPL iff for each arithmetical interpretation ƒ, ƒ(A) is provable in PA. Various modifications of this result are considered.
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