Dual Completeness of the Base of Peano and Gentzen Arithmetic

Васил Пенчев ИДЕЯТА ЗА ДУАЛНА ПЪЛНОТА ВЪЗ ОСНОВА НА ПЕАНОВА И ГЕНЦЕНОВА АРИТМЕТИКА Abstract: Peano arithmetic (including the principle of complete induction) can found infinity until being the metatheory or metalanguage of Gentzen arithmetic comprising... More

Васил Пенчев ИДЕЯТА ЗА ДУАЛНА ПЪЛНОТА ВЪЗ ОСНОВА НА ПЕАНОВА И ГЕНЦЕНОВА АРИТМЕТИКА Abstract: Peano arithmetic (including the principle of complete induction) can found infinity until being the metatheory or metalanguage of Gentzen arithmetic comprising both of the finite numbers and infinite ordinals (less than ). In its turn, Gentzen arithmetic (i. e. Peano axiomatics, in which complete induction is generalized to transfinite induction until ) can found Peano arithmetic being the metatheory or metalanguge of it. Peano arithmetic and Gentzen arithmetic have almost the same axiomatics. Each of them can serve as metalanguage as object language of the other. They are correlative each to other in the manner of Skolem. Describing such a case, we may introduce the term of dual (dualistic, mutual) foundation (also «dual-foundation» instead of «self-foundation» as well as «dualreference» instead of «self-reference») in relation to the completeness of arithmetic. The common and mutually Less