On first-order theories with provability operatorстатья
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Аннотация:In this paper the modal operator “x is provable in Peano Arithmetic” is incorporated into first-order theories. A provability extension of a theory is defined. Presburger Arithmetic of addition, Skolem Arithmetic of multiplication, and some first-order theories of partial consistency statements are shown to remain decidable after natural provability extensions. It is also shown that natural provability extension of a decidable theory may be undecidable.