Bulletin (New Series) of the American Mathematical Society

The theory of recursive functions, approaching its centennial

Stephen C. Kleene

Source: Bull. Amer. Math. Soc. (N.S.) Volume 5, Number 1 (1981), 43-61.

Related Works:

Primary Subjects: 03D20
Secondary Subjects: 03D10, 03A05

Full-text: Open access

Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.bams/1183548220
Mathematical Reviews number (MathSciNet): MR614313
Zentralblatt MATH identifier: 0486.03023

References

Ackermann, Wilhelm, [1928] Zum Hilbertschen Aufbau der reellen Zahlen, Math. Ann. 99, 118-133.
Jahrbuch database (Zbl): 54.0056.06
Mathematical Reviews (MathSciNet): MR1512441
Cantor, Georg, [1874] Über eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen, J. Reine Angew. Math. 77, 258-262.
Jahrbuch database (Zbl): 06.0057.01
Cantor, Georg, [1895-7] Beiträge zur Begründung der transfiniten Mengenlehre, Math. Ann. 46 (1895), 481-512; 49 (1897), 207-246.
Jahrbuch database (Zbl): 26.0081.01
Mathematical Reviews (MathSciNet): MR1510964
Church, Alonzo, [1932] A set of postulates for the foundation of logic, Ann. of Math. (2) 33, 346-366.
Zentralblatt MATH: 0004.14507
Mathematical Reviews (MathSciNet): MR1503059
Church, Alonzo, [1933] A set of postulates for the foundation of logic (second paper), Ann. of Math. (2) 34, 839-864.
Zentralblatt MATH: 0008.28902
Mathematical Reviews (MathSciNet): MR1503136
Church, Alonzo, [1936] An unsolvable problem of elementary number theory, Amer. J. Math. 58, 345-363.
Zentralblatt MATH: 0014.09802
Mathematical Reviews (MathSciNet): MR1507159
Church, Alonzo, [1936a] A note on the Entscheidungsproblem, J. Symbolic Logic 1, 40-41; Correction, 101-102.
Jahrbuch database (Zbl): 62.1058.05
Mathematical Reviews (MathSciNet): MR441682
Dedekind, Richard, [1888] Was sind und was sollen die Zahlen? Braunschweig.
Jahrbuch database (Zbl): 20.0049.05
Gödel, Kurt, [1931] Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme. I, Monatsh. Math. Phys. 38, 173-198.
Zentralblatt MATH: 0002.00101
Mathematical Reviews (MathSciNet): MR1549910
Gödel, Kurt, [1934] On undecidable propositions of formal mathematical systems, mimeographed notes by S. C. Kleene and J. B. Rosser on lectures at the Institute for Advanced Study, 1934; reprinted in Martin Davis, The undecidable, basic papers on undecidable propositions, unsolvable problems and computable functions, Raven Press, Hewlett, N. Y., 1965, pp. 39-74.
Zentralblatt MATH: 1099.03002
Mathematical Reviews (MathSciNet): MR2070909
Hilbert, David, [1918] Axiomatisches Denken, Math. Ann. 78, 405-415.
Jahrbuch database (Zbl): 46.0062.03
Mathematical Reviews (MathSciNet): MR1511909
Hilbert, David, [1926] Über das Unendliche, Math. Ann. 95, 161-190.
Jahrbuch database (Zbl): 51.0044.02
Mathematical Reviews (MathSciNet): MR1512272
Kleene, Stephen, C., [1935] A theory of positive integers in formal logic, Amer. J. Math. 57, 153-173; 219-244.
Zentralblatt MATH: 0011.00202
Kleene, Stephen, C., [1936] General recursive functions of natural numbers, Math. Ann. 112, 727-742.
Zentralblatt MATH: 0014.19402
Mathematical Reviews (MathSciNet): MR1513071
Kleene, Stephen, C., [1936a] λ-definability and recursiveness, Duke Math. J. 2, 340-353.
Zentralblatt MATH: 0014.38505
Mathematical Reviews (MathSciNet): MR1545927
Kleene, Stephen, C., [ 1938] On notation for ordinal numbers, J. Symbolic Logic 3, 150-155.
Jahrbuch database (Zbl): 64.0932.03
Kleene, Stephen, C., [1943] Recursive predicates and quantifiers, Trans. Amer. Math. Soc. 53, 41-73.
Zentralblatt MATH: 0063.03259
Mathematical Reviews (MathSciNet): MR7371
Kleene, Stephen, C., [1950] Recursive functions and intuitionistic mathematics, Proc. Internat. Congr. Math. (Cambridge, Mass., U.S.A., 1950), Amer Math. Soc., Providence, R. I., 1952, I, pp. 679-685.
Zentralblatt MATH: 0049.15001
Mathematical Reviews (MathSciNet): MR44468
Kleene, Stephen, C., [1952] Introduction to metamathematics, North-Holland, Amsterdam, P. Noordhoff, Groningen and D. Van Nostrand, Toronto and New York.
Mathematical Reviews (MathSciNet): MR51790
Kleene, Stephen, C., [1955] Arithmetical predicates and function quantifiers, Trans. Amer. Math. Soc. 79, 312-340.
Zentralblatt MATH: 0066.25703
Mathematical Reviews (MathSciNet): MR70594
Kleene, Stephen, C., [1955a] On the forms of the predicates in the theory of constructive ordinals (second paper), Amer. J. Math. 77, 405-428.
Zentralblatt MATH: 0067.25203
Mathematical Reviews (MathSciNet): MR70595
Kleene, Stephen, C., [1955b] Hierarchies of number-theoretic predicates, Bull. Amer. Math. Soc. 61, 193-213.
Zentralblatt MATH: 0066.25901
Mathematical Reviews (MathSciNet): MR70593
Kleene, Stephen, C., [1957] Mathematics, foundations of, Encyclopaedia Britannica, 1957 and subsequent printings. Beginning with 1974, the article was reworked to replace Kleene's section on intuitionism by a fuller treatment by Solomon Fefermann.
Kleene, Stephen, C., [1958] Mathematical logic: constructive and non-constructive operations, Proc. Internat. Congr. Math. (Edinburgh, 1958) (J. A. Todd, ed.), Cambridge at the University Press, 1960, pp. 137-153.
Zentralblatt MATH: 0126.01903
Mathematical Reviews (MathSciNet): MR114750
Kleene, Stephen, C., [1959] Recursive functionals and quantifiers of finite types. I, Trans. Amer. Math. Soc. 91, 1-52.
Zentralblatt MATH: 0088.01301
Mathematical Reviews (MathSciNet): MR102480
Kleene, Stephen, C., [1963] Recursive functionals and quantifiers of finite types. II, Trans. Amer. Math. Soc. 108, 106-142.
Zentralblatt MATH: 0121.01302
Mathematical Reviews (MathSciNet): MR153557
Kleene, Stephen, C., [1964] Computability, The Voice of America Forum Lectures, Phil. of Science Series, no. 6; reprinted, Philosophy of Science Today (Sidney Morgenbesser, ed.), Basic Books, New York and London, 1967, pp. 36-45.
Kleene, Stephen, C., [1967] Mathematical logic, Wiley, New York, London and Sidney.
Zentralblatt MATH: 0149.24309
Mathematical Reviews (MathSciNet): MR216930
Kleene, Stephen, C., [1969] The new logic, Sigma-Xi-RESA National Lecture, Southeast Tour (Spring 1969), Amer. Sci. 57, 333-347.
Kleene, Stephen, C., [1978] Recursive functionals and quantifiers of finite types revisited. I. Generalized Recursion Theory. II, Proc. 1977 Oslo Sympos. (J. E. Fenstad, R. O. Gandy and G. E.Sacks, eds.), North-Holland, Amsterdam, New York and Oxford, pp. 185-222.
Mathematical Reviews (MathSciNet): MR516936
Kleene, Stephen, C., [1980] Recursive functionals and quantifiers of finite types revisited. II, The Kleene Symposium (Madison, Wis., June 1978) (J. Barwise, H. J. Keisler and K. Kunen, eds.), North-Holland, Ansterdam, New York and Oxford, pp. 1-29.
Zentralblatt MATH: 0453.03048
Mathematical Reviews (MathSciNet): MR591873
Kleene, Stephen, C., [1981a] Origins of recursive function theory, 20th Annual Sympos. Foundations of Computer Science (San Juan, Puerto Rico, 1979), Institute of Electrical and Electronics Engineers (IEEE 79CH 1471-2C), pp. 371-382; Annals of the History of Computing, 1981 (to appear).
Zentralblatt MATH: 0998.03501
Mathematical Reviews (MathSciNet): MR598119
Kleene, Stephen, C., [1981b] Algorithms in various contexts, Proc. Sympos. Algorithms in Modern Mathematics and Computer Science (dedicated to Al-Khowarizmi) (Urgench, Khorezm Region, Uzbek, SSSR, 1979), Springer-Verlag, Berlin, Heidelberg and New York, 1981 (to appear).
Zentralblatt MATH: 0583.68007
Mathematical Reviews (MathSciNet): MR657625
Klein, Felix, [1908-9] Elementarmathematik vom höheren Standpunkte aus, Lithographed, Leipzig; English transl. from 3rd German ed. (E. R. Hedrick and C. A. Noble, eds.), Macmillan, New York, 1932.
Kronecker, Leopold, [1886] The saying quoted in the third paragraph of the text was spoken in his lecture on 21 September 1886 before der Deutschen Naturforscherversammlung zu Berlin, according to H. Weber, Leopold Kronecker, Math. Ann. 43 (1893), 1-25 (cf. p. 15). Also cf. Leopold Kronecker's Werke, hrg. K. Henzel, vol. 3, pt. 2, p. 203.
Löwenheim, Leopold, [1915] Über Möglichkeiten im Relativkalkül, Math. Ann. 76, 447-470.
Jahrbuch database (Zbl): 44.0078.01
Mathematical Reviews (MathSciNet): MR1511835
Marcov, A. A., [1951] Theory of algorithms, Amer. Math. Soc. Transl. (2) 15 (1960), 1-14.
Mathematical Reviews (MathSciNet): MR114753
Marcov, A. A., [1954] Theory of algorithms, The National Science Foundation, Washington, D.C., the Department of Commerce, U.S.A., and the Israel Program for Scientific Translation, Jerusalem, 1961.
Mathematical Reviews (MathSciNet): MR132690
Peano, Guiseppe, [1889] Arithmetices principia, nova methodo exposita, Turin, Bocca.
Jahrbuch database (Zbl): 21.0051.02
Peano, Guiseppe, [1891] Sul concetto di numero, Rivista di Matematica 1, 87-102; 256-267.
Jahrbuch database (Zbl): 23.0052.01
Péter, Rózsa, [1934] Über den Zusammenhang der verschiedenen Begriffe der rekursiven Funktion, Math. Ann. 110, 612-632.
Zentralblatt MATH: 0010.24101
Mathematical Reviews (MathSciNet): MR1512957
Péter, Rózsa, [1935] Konstruktion nichtrekursiver Funktionen, Math. Ann. 111, 42-60.
Zentralblatt MATH: 0011.00303
Mathematical Reviews (MathSciNet): MR1512975
Péter, Rózsa, [1935] Über die mehrfache Rekursion, Math. Ann. 113, 489-527.
Zentralblatt MATH: 0015.33902
Mathematical Reviews (MathSciNet): MR1513105
Post, Emil L., [1936] Finite combinatory processes-formulation 1, J. Symbolic Logic 1, 103-105.
Jahrbuch database (Zbl): 62.1060.01
Mathematical Reviews (MathSciNet): MR20527
Post, Emil L., [1943] Formal reductions of the general combinatorial decision problem, Amer. J. Math. 65, 197-215.
Zentralblatt MATH: 0063.06327
Mathematical Reviews (MathSciNet): MR7893
Post, Emil L., [1944] Recursively enumerable sets of positive integers and their decision problems, Bull. Amer. Math. Soc. 50, 284-316.
Zentralblatt MATH: 0063.06328
Mathematical Reviews (MathSciNet): MR10514
Post, Emil L., [1948] Degrees of recursive unsolvability, abstract (preliminary report) in Bull. Amer. Math. Soc. 54, 641-642.
Schröder, Ernst, [1895] Vorlesungen über die Algebra der Logik (exakte Logik), 3 Algebra und Logik der Relative, part 1, Leipzig.
Jahrbuch database (Zbl): 26.0074.01
Skolem, Thoralf, [1923] Begründung der elementaren Arithmetik durch die rekurrierende Denkweise ohne Anwendung scheinbare Veränderlichen mit unendlichem Ausdehnungsbereich, Skrifter utgit av Videnskapsselskapet i Kristiania, I. Matematisk-Naturvidenskabelig Klasse 1923, no. 6; English transl., Jean van Heijenoort, From Frege to Gödel, a source book in mathematical logic, 1879-1931, Harvard Univ. Press, Cambridge, Mass., 1967, pp. 302-333.
Smullyan, Raymond M., [1961] Theory of formal systems, Annals of Math. Studies, no. 47, Princeton Univ. Press, Princeton, N. J.
Zentralblatt MATH: 0097.24503
Mathematical Reviews (MathSciNet): MR152429
Turing, Allan Mathison, [1936-7] On computable numbers, with an application to the Entscheidungsproblem, Proc. London Math. Soc. (2) 42, 230-265; A Correction, 43 (1937), 544-546.
Jahrbuch database (Zbl): 63.0823.02
Turing, Allan Mathison, [1939] Systems of logic based on ordinals, Proc. London Math. Soc. (2) 45, 161-228.
Jahrbuch database (Zbl): 65.1102.02

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Bulletin (New Series) of the American Mathematical Society

Bulletin (New Series) of the American Mathematical Society