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Kleene's Amazing Second Recursion Theorem

Published online by Cambridge University Press:  15 January 2014

Yiannis N. Moschovakis
Yiannis N. Moschovakis*
Affiliation:
Department of Mathematics, University of California, Los Angeles, CA 90095-1555, USA Department of Mathematics, University of Athens, Greece, E-mail:, [email protected]
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Extract

This little gem is stated unbilled and proved (completely) in the last two lines of §2 of the short note Kleene [1938]. In modern notation, with all the hypotheses stated explicitly and in a strong (uniform) form, it reads as follows:

Second Recursion Theorem (SRT). Fix a set V ⊆ ℕ, and suppose that for each natural number n ϵ ℕ = {0, 1, 2, …}, φn: ℕ1+n ⇀ V is a recursive partial function of (1 + n) arguments with values in V so that the standard assumptions (a) and (b) hold with

.

(a) Every n-ary recursive partial function with values in V is for some e.

(b) For all m, n, there is a recursive function : Nm+1 → ℕ such that

.

Then, for every recursive, partial function f of (1+m+n) arguments with values in V, there is a total recursive function of m arguments such that

Proof. Fix e ϵ ℕ such that and let .

We will abuse notation and write ž; rather than ž() when m = 0, so that (1) takes the simpler form

in this case (and the proof sets ž = S(e, e)).

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 16 , Issue 2 , June 2010 , pp. 189 - 239
Copyright
Copyright © Association for Symbolic Logic 2010

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References

REFERENCES

[1954] Addison, J. W., On some points of the theory of recursive functions, Ph.D. thesis, University of Wisconsin, 1954.Google Scholar
[1959] Addison, J. W., Separation principles in the hierarchies of classical and effective descriptive set theory, Fundamenta Mathematicae, vol. 46 (1959), pp. 123–135.Google Scholar
[1974] Boolos, George S., Burgess, John P., and Jeffrey, Richard C., Computability and logic, Cambridge University Press, 1974.Google Scholar
[1962] Ceitin, G., Algorithmic operators in constructive metric spaces, Trudy Matematicheskogo Instituta imeni V. A. Steklova, vol. 67 (1962), pp. 295–361.Google Scholar
[1950a] Davis, Martin, On the theory of recursive unsolvability, Ph.D. thesis, Princeton University, 1950.Google Scholar
[1950b] Davis, Martin, Relatively recursive functions and the extended Kleene hierarchy, Proceedings of the International Congress of Mathematicians, Cambridge, Massachusetts, 1950, 1950, p. 723.Google Scholar
[1987] Debs, Gabriel, Effective properties in compact sets of Borel functions, Mathematika, vol. 34 (1987), pp. 64–68.Google Scholar
[2009] Debs, Gabriel, Borel extractions of converging sequences in compact sets of Borel functions, Journal of Mathematical Analysis and Applications, vol. 350 (2009), pp. 731–744.Google Scholar
[1958] Friedberg, Richard M., Un contre-example relatif aux fonctionnelles récursives, Comptes Rendus de l'Académie des Sciences, Paris, vol. 247 (1958), pp. 852–854.Google Scholar
[2009] Gregoriades, Vassilis, Effective descriptive set theory and applications in analysis, Ph.D. thesis, University of Athens, 2009.Google Scholar
[1989] Jackson, S., AD and the very fine structure of L(ℝ), Bulletin of the American Mathematical Society, vol. 21 (1989), pp. 77–81.Google Scholar
[1977] Kechris, A. S. and Moschovakis, Y. N., Recursion in higher types, Handbook of mathematical logic (Barwise, J., editor), Studies in Logic,No. 90, North Holland, Amsterdam, 1977, pp. 681–737.Google Scholar
[1935] Kleene, Stephen C., General recursive functions of natural numbers, Mathematische Annalen, vol. 112 (1935), pp. 727–742.Google Scholar
[1938] Kleene, Stephen C., On notation for ordinal numbers, The Journal of Symbolic Logic, vol. 3 (1938), pp. 150–155.Google Scholar
[1943] Kleene, Stephen C., Recursive predicates and quantifiers, Transactions of the American Mathematical Society, vol. 53 (1943), pp. 41–73.Google Scholar
[1944] Kleene, Stephen C., On the form of predicates in the theory of constructive ordinals, American Journal of Mathematics, vol. 66 (1944), pp. 41–58.CrossRefGoogle Scholar
[1945] Kleene, Stephen C., On the interpretation of intuitionistic number theory, The Journal of Symbolic Logic, vol. 10 (1945), pp. 109–124.CrossRefGoogle Scholar
[1952] Kleene, Stephen C., Introduction to metamathematics, D. Van Nostrand Company, North Holland Company, 1952.Google Scholar
[1955a] Kleene, Stephen C., Arithmetical predicates and function quantifiers, Transactions of the American Mathematical Society, vol. 79 (1955), pp. 312–340.Google Scholar
[1955b] Kleene, Stephen C., On the form of predicates in the theory of constructive ordinals (second paper), American Journal of Mathematics, vol. 77 (1955), pp. 405–428.Google Scholar
[1955c] Kleene, Stephen C., Hierarchies of number theoretic predicates, Bulletin of the American Mathematical Society, vol. 61 (1955), pp. 193–213.Google Scholar
[1959a] Kleene, Stephen C., Quantification of number-theoretic functions, Compositio Mathematica, vol. 14 (1959), pp. 23–40.Google Scholar
[1959b] Kleene, Stephen C., Recursive functionals and quantifiers of finite types I, Transactions of the American Mathematical Society, vol. 91 (1959), pp. 1–52.Google Scholar
[1965] Kleene, Stephen C. and Vesley, Richard Eugene, The foundations of intuitionistic mathematics, North Holland, Amsterdam, 1965.Google Scholar
[1959] Kreisel, Georg, Analysis of Cantor-Bendixson Theorem by means of the analytic hierarchy, Bulletin de l'Academie Polonaise des Sciences, Série des sciences mathématiques, astronomiques et physiques, vol. 7 (1959), pp. 621–626.Google Scholar
[1957] Kreisel, Georg, Lacombe, Daniel, and Shoenfield, Joseph R., Partial recursive functionals and effective operations, Constructivity in mathematics: proceedings of the colloquium held in Amsterdam, 1957 (Heyting, A., editor), North Holland Publishing Company, 1957, pp. 195–207.Google Scholar
[1966] Kuratowski, K., Topology, vol. 1, Academic Press, New York and London, 1966, translated from the French Topologie, vol. 1, PWN, Warsaw, 1958.Google Scholar
[1905] Lebesgue, Henri, Sur les fonctions represéntables analytiquement, Journal de Mathématiques 6e serie, vol. 1 (1905), pp. 139–216.Google Scholar
[2002] Margenstern, Maurice and Rogozhin, Yurii, Self-describing Turing machines, Fundamenta Informatica, vol. 50 (2002), no. 3, pp. 285–303.Google Scholar
[1954] Markwald, Werner, Zur Theorie der konstruktiven Wohlordnungen, Mathematische Annalen, vol. 127 (1954), pp. 135–149.Google Scholar
[1988] Martin, D. A. and Steel, J. R., Projective determinacy, Proceedings of the National Academy of Sciences of the United States of America, vol. 85 (1988), no. 18, pp. 6582–6586.CrossRefGoogle ScholarPubMed
[1966] Moschovakis, Yiannis N., Many-one degrees of the predicates Ha(x), Pacific Journal of Mathematics, vol. 18 (1966), pp. 329–342.Google Scholar
[1970] Moschovakis, Yiannis N., Determinacy and prewellorderings of the continuum, Mathematical logic and foundations of set theory (Bar-Hillel, Y., editor), North-Holland, Amsterdam, 1970, pp. 24–62.Google Scholar
[1974] Moschovakis, Yiannis N., Structural characterizations of classes of relations, Generalized recursion theory (Fenstad, J. E. and Hinman, P. G., editors), North-Holland, Amsterdam, 1974, pp. 53–81.Google Scholar
[2009a] Moschovakis, Yiannis N., Descriptive set theory, second edition, Mathematical Surveys and Monographs., vol. 155, American Mathematical Society, 2009.Google Scholar
[2009b] Moschovakis, Yiannis N., Kleen's amazing second recursion theorem (extended abstract), CSL 2009 (Grädel, E. and Kahle, R., editors), Lecture Notes in Computer Science, no. 5771, Springer-Verlag, Berlin - Heidelberg, 2009, pp. 24–39.Google Scholar
[2010] Moschovakis, Yiannis N., Classical descriptive set theory as a refinement of effective descriptive set theory, (2010), submitted.CrossRefGoogle Scholar
[1946] Mostowski, A., On definable sets of positive integers, Fundamenta Mathematicae, vol. 34 (1946), pp. 81–112.Google Scholar
[1951] Mostowski, A., A classification of logical systems, Studia Philosophica, vol. 4 (1951), pp. 237–274.Google Scholar
[1955] Myhill, John, Creative sets, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 1 (1955), pp. 97–108.Google Scholar
[1955] Myhill, John and Shepherdson, John C., Effective operations on partial recursive functions, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 1 (1955), pp. 310–317.Google Scholar
[1947] Nelson, David, Recursive functions and intuitionistic number theory, Transactions of the American Mathematical Society, vol. 61 (1947), pp. 308–368.Google Scholar
[1974] Nelson, George C., Many-one reducibility within the Turing degrees of the hyperarithmetic sets Ha(x), Transactions of the American Mathematical Society, vol. 191 (1974), pp. 1–44.Google Scholar
[1944] Post, Emil L., Recursively enumerable sets of positive integers and their decision problems, Bulletin of the American Mathematical Society, vol. 50 (1944), pp. 284–316.Google Scholar
[1967] Pour-El, Marian Boykan and Kripke, Saul, Deduction-preserving “recursive isomorphism” between theories, Fundamenta Mathematicae, vol. 61 (1967), pp. 141–163.Google Scholar
[1950] Robinson, Raphael M., An essentially undecidable axiom system, Proceedings of the International Congress of Mathematics, 1950, pp. 729–730.Google Scholar
[1967] Rogers, Hartley Jr., Theory of recursive functions and effective computability, McGraw-Hill, 1967.Google Scholar
[1990] Sacks, Gerald E., Higher recursion theory, Perspectives in Mathematical Logic, Springer, 1990.Google Scholar
[1976] Solovay, Robert M., Provability interpretations of modal logic, Israel Journal of Mathematics, vol. 25 (1976), pp. 287–304.Google Scholar
[1955] Spector, Clifford, Recursive wellorderings, The Journal of Symbolic Logic, vol. 20 (1955), pp. 151–163.Google Scholar
[1917] Suslin, M., Sur une definition des ensembles measurables B sans nombres transfinis, Comptes Rendus de l'Académie des Sciences, Paris, vol. 164 (1917), pp. 88–91.Google Scholar
[1958] Wang, Hao, Alternative proof of a theorem of Kleene, The Journal of Symbolic Logic, vol. 23 (1958), p. 250.Google Scholar
[1988] Woodin, W. H., Supercompact cardinals, sets of reals, and weakly homogeneous trees, Proceedings of the National Academy of Sciences of the United States of America, vol. 85 (1988), pp. 6587–6591.Google Scholar