Teorema ketaklengkapan Gödel: Perbedaan antara revisi
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'''Teorema ketaklengkapan Gödel''' ({{lang-en|Gödel's incompleteness theorems}}) adalah dua [[teorema]] [[logika matematika]] yang menetapkan batasan (''limitation'') inheren dari semua kecuali [[:en:axiomatic system|sistem aksiomatik]] yang paling trivial yang mampu mengerjakan [[aritmetika]]. Teorema-teorema ini, dibuktikan oleh [[Kurt Gödel]] pada tahun 1931, penting baik dalam logika matematika maupun dalam [[filsafat matematika]]. Kedua hasil ini secara luas, tetapi tidak secara universal, ditafsirkan telah menunjukkan bahwa [[:en:Hilbert's program|program Hilbert]] untuk menghitung himpunan lengkap dan konsisten dari [[:en:axiom|aksioma-aksioma]] bagi semua [[matematika]] adalah tidak mungkin, sehingga memberikan jawaban negatif terhadap [[:en:Hilbert's second problem|soal Hilbert yang kedua]].
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Now comes the trick: The notion of provability itself can also be encoded by Gödel numbers, in the following way. Since a proof is a list of statements which obey certain rules, the Gödel number of a proof can be defined. Now, for every statement ''p'', one may ask whether a number ''x'' is the Gödel number of its proof. The relation between the Gödel number of ''p'' and ''x'', the potential Gödel number of its proof, is an arithmetical relation between two numbers. Therefore there is a statement form Bew(''y'') that uses this arithmetical relation to state that a Gödel number of a proof of ''y'' exists:
:Bew(''y'') = ∃ ''x'' (
The name '''Bew''' is short for ''beweisbar'', the German word for "provable"; this name was originally used by Gödel to denote the provability formula just described. Note that "Bew(''y'')" is merely an abbreviation that represents a particular, very long, formula in the original language of ''T''; the string "Bew" itself is not claimed to be part of this language.
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* [[Ernest Nagel]], James Roy Newman, Douglas Hofstadter, 2002 (1958). ''Gödel's Proof'', revised ed. ISBN 0-8147-5816-9. {{MathSciNet|2002i:03001}}
* [[Rudy Rucker]], 1995 (1982). ''Infinity and the Mind: The Science and Philosophy of the Infinite''. Princeton Univ. Press. {{MathSciNet|84d:03012}}
* Smith, Peter, 2007. ''[http://www.godelbook.net/ An Introduction to Gödel's Theorems.] {{Webarchive|url=https://web.archive.org/web/20051023200804/http://www.godelbook.net/ |date=2005-10-23 }}'' Cambridge University Press. [http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=AUCN&pg6=PC&pg7=ALLF&pg8=ET&s4=Smith%2C%20Peter&s5=&s6=&s7=&s8=All&yearRangeFirst=&yearRangeSecond=&yrop=eq&r=2&mx-pid=2384958 MathSciNet]{{Pranala mati|date=Januari 2023 |bot=InternetArchiveBot |fix-attempted=yes }}
* N. Shankar, 1994. ''Metamathematics, Machines and Gödel's Proof'', Volume 38 of Cambridge tracts in theoretical computer science. ISBN 0-521-58533-3
* [[Raymond Smullyan]], 1991. ''Godel's Incompleteness Theorems''. Oxford Univ. Press.
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