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Binumerability in a sequence of theories

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Binumerability in a sequence of theories RENDICONTI del SEMINARIO MATEMATICO della UNIVERSITÀ DI PADOVA FRANCO PARLAMENTO Binumerability in a sequence of theories Rendiconti del Seminario Matematico della Università di Padova, tome 65 (1981), p. 9-12. <http://www.numdam.org/item?id=RSMUP_1981__65__9_0> © Rendiconti del Seminario Matematico della Università di Padova, 1981, tous droits réservés. L’accès aux archives de la revue « Rendiconti del Seminario Matematico della Università di Padova » (http://rendiconti.math.unipd.it/) implique l’ac- cord avec les conditions générales d’utilisation (http://www.numdam.org/legal. php). Toute utilisation commerciale ou impression systématique est consti- tutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Binumerability in a Sequence of Theories. FRANCO PARLAMENTO (*) SUMMARY - In this note we answer a question raised by A. Ursini in [2]. In that work he defines a denumerable sequence of arithmetic theories Qn , whose union is complete, and asks a question concerning the binumer- ability of the relations in Qn . We show that a relation is in if and only if it is binumerable in Qn. We are going to follow the notations and terminology of [1] and [2]. In particular Ko is the language of first order arithmetic, and a K-system is a set of sentences in the language .K. y) is the relation y is a proof of x from axioms in T ». Let’s recall that, given a K-system, where Ko c _K, a numerical relation con is called nu- merable in T is there is a formula ... , in .g such that In that case we say that q numerates R in T. A relation R is binumerable in T if there exists a formula p in K such that p numerates numerates con - .R. The result expressed in the following proposition is obtained as a straigtforward application of the « Rosser trick )}

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