Geodesic
G\"odel incompleteness theorems and the limits of their applicability.~I
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 65 (2010) no. 5, pp. 857-899

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This is a survey of results related to the Gödel incompleteness theorems and the limits of their applicability. The first part of the paper discusses Gödel's own formulations along with modern strengthenings of the first incompleteness theorem. Various forms and proofs of this theorem are compared. Incompleteness results related to algorithmic problems and mathematically natural examples of unprovable statements are discussed. Bibliography: 68 titles.
Keywords: Gödel theorems, incompleteness, proof, computability. 
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     title = {G\"odel incompleteness theorems and the limits of their {applicability.~I}},
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L. D. Beklemishev. G\"odel incompleteness theorems and the limits of their applicability.~I. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 65 (2010) no. 5, pp. 857-899. http://geodesic.mathdoc.fr/item/RM_2010_65_5_a2/