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== Generalized theorem ==
In the context of his theory of numberings, [[Yury Yershov|Ershov]] showed that Kleene's recursion theorem holds for any [[precomplete numbering]] ([[#CITEREFBarendregtTerwijn2019|Barendregt & Terwijn 2019]]: p. 1151). A Gödel numbering is a precomplete numbering on the set of computable functions so the generalized theorem yields the Kleene recursion theorem as a special case (for an English survey see [[#CITEREFErshov1999|Ershov 1999]]: pp§4. 473-50314).
Given a precomplete numbering <math>\nu</math> then for any partial computable function <math>f</math> with two parameters there exists a total computable function <math>t</math> with one parameter such that
== References ==
* {{Cite journal|last1=Barendregt|first1=Henk|author-link1=Henk Barendregt|last2=Terwijn|first2=Sebastiaan A.|date=1 October 2019|title=Fixed point theorems for precomplete numberings|url=http://www.sciencedirect.com/science/article/pii/S016800721930048X|journal=Annals of Pure and Applied Logic|language=English|volume=170|issue=10|pages=1151-{{ndash}}1161|doi=10.1016/j.apal.2019.04.013|issn=0168-0072|access-date=6 May 2020|url-status=live|url-access=subscription}}
* {{Cite book|last=Cutland,|first=Nigel N.J.,|author-link=Nigel Cutland|date=1980, ''|title=Computability: An introductionIntroduction to recursiveRecursive function theory'',Function Theory|url=https://books.google.com/books?id=wAstOUE36kcC&printsec=frontcover|publisher=[[Cambridge University Press]]|language=English|doi=10. {{isbn1017/cbo9781139171496|isbn=0-521-29465-7|oclc=488175597|access-date=6 May 2020|url-status=live}}
* {{Cite book|last=Ershov|first=Yuri L|author-link=Yury Yershov|editor-last=Griffor|editor-first=Edward R|url=https://books.google.com/books?id=KqeXZ4pPd5QC&printsec=frontcover|title=Handbook of Computability Theory|chapter=Part 4: Mathematics and Computability Theory. 14. Theory of numbering|series=Studies in logic and the foundations of mathemtics|date=1999|volume=140|pages=473-503|publisher=[[Elsevier]]||___location=Amsterdam|language=English|oclc=162130533|isbn=978-0-444-89882-1|___locationaccess-date=Amsterdam6 May 2020|languageurl-status=English|oclc=162130533live}}
* {{Cite [[Stephen Cole Kleenejournal|last=Kleene, |first=S. C.]],|author-link=Stephen Cole Kleene|date=1938,|title=On "[notation for ordinal numbers|url=http://www.thatmarcusfamily.org/philosophy/Course_Websites/Readings/Kleene%20-%20Ordinals.pdf On Notation for Ordinal Numbers]", |journal=[[Journal of Symbolic Logic]] |language=English|volume=3, 150–155|issue=4|pages=150{{ndash}}155|doi=10.2307/2267778|issn=0022-4812|access-date=6 May 2020|url-status=live}}
* {{Cite book|last=Kleene|first=S. C.|author-link=Stephen Cole Kleene|date=1952|title=Introduction to Metamathematics|url=https://archive.org/details/BubliothecaMathematicaStephenColeKleeneIntroductionToMetamathematicsWoltersNoordhoffPublishing1971|publisher=[[North-Holland Publishing]]|language=English|series=Bibliotheca Mathematica|isbn=9780720421033|oclc=459805591|access-date=6 May 2020|url-status=live}}
* Kleene, S.C., 1952, ''Introduction to Metamathematics'', North-Holland. {{isbn|0-7204-2103-9}}
* [[Carl{{Cite journal|last=Jockusch|Jockusch, first=C. G.|author-link1=Carl Jr.]]; Jockusch|last2=Lerman, |first2=M.; |last3=Soare, |first3=R. I.;|author-link3=Robert andI. Soare|last4=Solovay, |first4=R. M.,|author-link4=Robert 1989,M. "Solovay|date=1989|title=Recursively enumerable sets modulo iterated jumps and extensions of Arslanov's completeness criterion", |journal=[[The Journal of Symbolic Logic]], |language=English|volume=54|issue=4, |pages=1288&{{ndash;}}1323|doi=10. 1017/S0022481200041104|issn=0022-4812}}
* {{Cite book|last=Jones,|first=Neil N.D.J., |date=1997,|author-link=Neil ''D. Jones|title=Computability and Complexitycomplexity: fromFrom a programming perspective'',Programming Perspective|publisher=[[MIT Press. {{isbn]]|isbn=978-0-262-10064-9|___location=Cambridge, Massachusetts|language=English|oclc=981293265}}
* {{Cite book|last=Rogers|first=Hartley|author-link=Hartley Rogers Jr.|date=1967|title=Theory of recursive functions and effective computability|url=https://archive.org/details/theoryofrecursiv00roge|publisher=[[MIT Press]]|___location=Cambridge, Massachusetts|language=English|oclc=933975989|isbn=9780262680523|access-date=6 May 2020|url-status=live|url-access=registration}}
* Rogers, H., 1967, ''Theory of Recursive Functions and Effective Computability'', MIT Press. {{isbn|0-262-68052-1}}; {{isbn|0-07-053522-1}}
* Soare,{{Cite book|last=Soare|first=R. I.,|author-link=Robert 1987,I. ''Soare|title=Recursively enumerableEnumerable setsSets and degrees'',Degrees: A Study of Computable Functions and Computably Generated Sets|series=Perspectives in Mathematical Logic, |date=1987|publisher=[[Springer-Verlag. {{]]|isbn|3=978-5400-387-15299-78|___location=Berlin; New York|language=English|oclc=318368332}}.
== External links ==
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