Geodesic
On arithmetic with the notion of ``attainable number''
Izvestiya. Mathematics , Tome 29 (1987) no. 3, pp. 477-510.

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The author investigates the extension of formal arithmetic by the proposition that a concrete “large” natural number is not attainable. It is shown in the article that although the resulting system is inconsistent, the only formulas in the language of arithmetic which can be derived by “short” proofs are those which are theorems of arithmetic. Bibliography: 10 titles. 
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E. S. Bozhich. On arithmetic with the notion of ``attainable number''. Izvestiya. Mathematics , Tome 29 (1987) no. 3, pp. 477-510. http://geodesic.mathdoc.fr/item/IM2_1987_29_3_a0/

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