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Author Levin, Leonid A.
Source ACM Digital Library
Content type Text
Publisher Association for Computing Machinery (ACM)
File Format PDF
Copyright Year ©2013
Language English
Subject Domain (in DDC) Computer science, information & general works ♦ Data processing & computer science
Subject Keyword Church's thesis ♦ Kolmogorov complexity ♦ Turing ♦ Incompleteness
Abstract Gödel Incompleteness Theorem leaves open a way around it, vaguely perceived for a long time but not clearly identified. (Thus, Gödel believed informal arguments can answer any math question.) Closing this loophole does not seem obvious and involves Kolmogorov complexity. (This is unrelated to, well studied before, complexity quantifications of the usual Gödel effects.) I consider extensions $\textit{U}$ of the universal partial recursive predicate (or, say, Peano Arithmetic). I prove that any $\textit{U}$ either leaves an $\textit{n}-bit$ input (statement) unresolved or contains nearly all information about the $\textit{n}-bit$ prefix of any r.e. real ρ (which is $\textit{n}$ bits for some ρ). I argue that creating significant information about a $\textit{specific}$ math sequence is impossible regardless of the methods used. Similar problems and answers apply to other unsolvability results for tasks allowing multiple solutions, for example, nonrecursive tilings.
ISSN 00045411
Age Range 18 to 22 years ♦ above 22 year
Educational Use Research
Education Level UG and PG
Learning Resource Type Article
Publisher Date 2013-05-03
Publisher Place New York
e-ISSN 1557735X
Journal Journal of the ACM (JACM)
Volume Number 60
Issue Number 2
Page Count 9
Starting Page 1
Ending Page 9

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Source: ACM Digital Library