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Kleene’s Amazing Second Recursion Theorem

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Computer Science Logic (CSL 2009)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5771))

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Abstract

This little gem is stated unbilled and proved (completely) in the last two lines of \(\S\)2 of the short note Kleene (1938).

Part of an expository article in preparation, written to commemorate the passage of 100 years since the birth of Stephen Cole Kleene.

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References

  • Addison, J.W.: On some points of the theory of recursive functions. Ph.D. Thesis, University of Wisconsin (1954)

    Google Scholar 

  • Addison, J.W.: Separation principles in the hierarchies of classical and effective descriptive set theory. Fundamenta Methematicae 46, 123–135 (1959)

    MathSciNet  MATH  Google Scholar 

  • Boolos, G.S., Burgess, J.P., Jeffrey, R.C.: Computability and logic. Cambridge University Press, Cambridge (1974)

    MATH  Google Scholar 

  • Ceitin, G.: Algorithmic operators in constructive metric spaces. Trudy Mat. Inst. Steklov. 67, 295–361 (1962)

    MathSciNet  Google Scholar 

  • Davis, M.: Relatively recursive functions and the extended Kleene hierarchy. In: Proceedings of the International Congress of Mathematicians, Cambridge, Mass, p. 723 (1950)

    Google Scholar 

  • Friedberg, R.M.: Un contre-example relatif aux fonctionnelles récursives. Comptes Rendus Acad. Sci. Paris 247, 852–854 (1958)

    MATH  Google Scholar 

  • Kleene, S.C.: General recursive functions of natural numbers. Mathematische Annalen 112, 727–742 (1935)

    Article  MathSciNet  MATH  Google Scholar 

  • Kleene, S.C.: On notation for ordinal numbers. Journal of Symbolic Logic 3, 150–155 (1938)

    Article  MATH  Google Scholar 

  • Kleene, S.C.: Recursive predicates and quantifiers. Transactions of the American Mathematical Society 53, 41–73 (1943)

    Article  MathSciNet  MATH  Google Scholar 

  • Kleene, S.C.: On the form of predicates in the theory of constructive ordinals. American Journal of Mathematics 66, 41–58 (1944)

    Article  MathSciNet  MATH  Google Scholar 

  • Kleene, S.C.: Introduction to metamathematics. D. Van Nostrand Co. / North Holland Co. (1952)

    Google Scholar 

  • Kleene, S.C.: Arithmetical predicates and function quantifiers. Transactions of the American Mathematical Society 79, 312–340 (1955)

    Article  MathSciNet  MATH  Google Scholar 

  • Kleene, S.C.: On the form of predicates in the theory of constructive ordinals (second paper). American Journal of Mathematics 77, 405–428 (1955)

    Article  MathSciNet  MATH  Google Scholar 

  • Kleene, S.C.: Hierarchies of number theoretic predicates. Bulletin of the American Mathematical Society 61, 193–213 (1955)

    Article  MathSciNet  MATH  Google Scholar 

  • Kleene, S.C.: Quantification of number-theoretic functions. Compositio Mathematica 14, 23–40 (1959)

    MathSciNet  MATH  Google Scholar 

  • Kreisel, G., Lacombe, D., Shoenfield, J.R.: Partial recursive functionals and effective operations. In: Heyting, A. (ed.) Constructivity in mathematics: proceedings of the colloquium held in Amsterdam, 1957, pp. 195–207. North Holland Publishing Co., Amsterdam (1957)

    Google Scholar 

  • Markwald, W.: Zur Theorie der konstruktiven Wohlordnungen. Mathematischen Annalen 127, 135–149 (1954)

    Article  MathSciNet  MATH  Google Scholar 

  • Martin, D.A., Steel, J.R.: Projective determinacy. Proceedings of the National Academy of Sciences, USA 85, 6582–6586 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  • Moschovakis, Y.N.: Determinacy and prewellorderings of the continuum. In: Bar-Hillel, Y. (ed.) Mathematical Logic and Foundations of Set Theory, pp. 24–62. North-Holland, Amsterdam (1970)

    Google Scholar 

  • Moschovakis, Y.N.: Descriptive set theory. Mathematical Surveys and Monographs, vol. 155. American Mathematical Society, Providence (2009)

    Google Scholar 

  • Moschovakis, Y.N.: Classical descriptive set theory as a refinement of effective descriptive set theory (to appear, 2010)

    Google Scholar 

  • Mostowski, A.: On definable sets of positive integers. Fundamenta Methematicae 34, 81–112 (1946)

    MathSciNet  MATH  Google Scholar 

  • Mostowski, A.: A classification of logical systems. Studia Philosophica 4, 237–274 (1951)

    MATH  Google Scholar 

  • Myhill, J.: Creative sets. Zeitschrifft für Mathematische Logik und Grundlagen der Mathematik 1, 97–108 (1955)

    Article  MathSciNet  MATH  Google Scholar 

  • Myhill, J., Shepherdson, J.C.: Effective operations on partial recursive functions. Zeitschrifft für Mathematische Logik und Grundlagen der Mathematik 1, 310–317 (1955)

    Article  MathSciNet  MATH  Google Scholar 

  • Post, E.L.: Recursively enumerable sets of positive integers and their decision problems. Bulletin of the American Mathematical Society 50, 284–316 (1944)

    Article  MathSciNet  MATH  Google Scholar 

  • Pour-El, M.B., Kripke, S.: Deduction-preserving “recursive isomorphisms” between theories. Fundamenta Mathematicae 61, 141–163 (1967)

    MathSciNet  MATH  Google Scholar 

  • Robinson, R.M.: An essentially undecidable axiom system. In: Proceedings of the International Congress of Mathematics, pp. 729–730 (1950)

    Google Scholar 

  • Rogers Jr., H.: Theory of recursive functions and effective computability. McGraw-Hill, New York (1967)

    MATH  Google Scholar 

  • Solovay, R.M.: Provability interpretations of modal logic. Israel Journal of Mathematics 25, 287–304 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  • Spector, C.: Recursive wellorderings. Journal of Symbolic Logic 20, 151–163 (1955)

    Article  MathSciNet  MATH  Google Scholar 

  • Suslin, M.: Sur une definition des ensembles measurables B sans nombres transfinis. Comptes Rendus Acad. Sci. Paris 164, 88–91 (1917)

    Google Scholar 

  • Woodin, W.H.: Supercompact cardinals, sets of reals, and weakly homogeneous trees. Proceedings of the National Academy of Sciences, USA 85, 6587–6591 (1988)

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of California, Los Angeles, CA, USA

    Yiannis N. Moschovakis

  2. Department of Mathematics, University of Athens, Greece

    Yiannis N. Moschovakis

Authors
  1. Yiannis N. Moschovakis
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Editors and Affiliations

  1. Mathematische Grundlagen der Informatik, RWTH Aachen, 52056, Aachen, Germany

    Erich Grädel

  2. CENTRIA and Departamento de Matemática, Universidade Nova de Lisboa, 2829-516, Caparica, Portugal

    Reinhard Kahle

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Moschovakis, Y.N. (2009). Kleene’s Amazing Second Recursion Theorem. In: Grädel, E., Kahle, R. (eds) Computer Science Logic. CSL 2009. Lecture Notes in Computer Science, vol 5771. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04027-6_5

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  • DOI: https://doi.org/10.1007/978-3-642-04027-6_5

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-04026-9

  • Online ISBN: 978-3-642-04027-6

  • eBook Packages: Computer ScienceComputer Science (R0)

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