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Generic Gödel's incompleteness theorem
A. N. Rybalovab a Sobolev Institute of Mathematics, Siberian Branch, Russian Academy of Sciences, ul. Pevtsova 13,
Omsk, 644099 Russia
b Omsk State Technical University, pr. Mira 11, Omsk, 644050 Russia
Abstract:
Gödel’s incompleteness theorem asserts that if formal arithmetic is consistent then there exists an arithmetic statement such that neither the statement nor its negation can be derived from the axioms of formal arithmetic. Previously [Sib. El. Mat. Izv., 12 (2015), 185–189], it was proved that formal arithmetic remains incomplete if, instead of the set of all arithmetic statements, we consider any set of some class of “almost all” statements (a class of socalled strongly generic subsets). This result is strengthened as follows: formal arithmetic is incomplete for any generic subset of arithmetic statements (i.e., a subset of asymptotic density 1).
Keywords:
Gödel’s theorem, formal arithmetic, generic subsets of arithmetic statements.
Received: 13.04.2016 Revised: 16.06.2016
Citation:
A. N. Rybalov, “Generic Gödel's incompleteness theorem”, Algebra Logika, 56:3 (2017), 348–353; Algebra and Logic, 56:3 (2017), 232–235
Linking options:
https://www.mathnet.ru/eng/al795 https://www.mathnet.ru/eng/al/v56/i3/p348
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Abstract page: | 258 | Full-text PDF : | 42 | References: | 37 | First page: | 10 |
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