Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Remote Access
Green Open Access
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Recursive functionals and quantifiers of finite types. I


Author: S. C. Kleene
Journal: Trans. Amer. Math. Soc. 91 (1959), 1-52
MSC: Primary 02.00
MathSciNet review: 0102480
Full-text PDF Free Access

References | Similar Articles | Additional Information

References [Enhancements On Off] (What's this?)

  • [1] L. E. J. Brouwer, Beweis, dass jede volle Funktion gleichmässig stetig ist, Nederl. Akad. Wetensch. vol. 27 (1924) pp. 189-193.
  • [2] -, Über Definitionsbereiche von Funktionen, Math. Ann. vol. 97 (1927) pp. 60-75. MR 1512354
  • [3] -, Points and spaces, Canad. J. Math. vol. 6 (1953) pp. 1-17.
  • [4] A. Church, An unsolvable problem of elementary number theory, Amer. J. Math. vol. 58 (1936) pp. 345-363. MR 1507159
  • [5] M. Davis, On the theory of recursive unsolvability, Ph.D. thesis (typewritten), Princeton University, 1950.
  • [6] A. Grzegorczyk, Some classes of recursive functions, Rozprawy Mat. no. 4, Warsaw, 1953, 46 pp. MR 0060426 (15:667d)
  • [7] -, Computable functionals, Fund. Math. vol. 42 (1955) pp. 168-202. MR 0086756 (19:238b)
  • [8] S. C. Kleene, General recursive functions of natural numbers, Math. Ann. vol. 112 (1936) pp. 727-742. MR 1513071
  • [9] -, A note on recursive functions, Bull. Amer. Math. Soc. vol. 42 (1936) pp. 544-546. MR 1563352
  • [10] -, Recursive predicates and quantifiers, Trans. Amer. Math. Soc. vol. 53 (1943) pp. 41-73. For an error in §15, cf. [15, §19]. MR 0007371 (4:126b)
  • [11] -, On the forms of the predicates in the theory of constructive ordinals, Amer. J. Math. vol. 66 (1944) pp. 41-58. An error is corrected in [15]. MR 0009757 (5:197e)
  • [12] -, Recursive functions and intuitionistic mathematics, Proceedings of the International Congress of Mathematicians (Cambridge, Mass. Aug. 30-Sept. 6, 1950) vol. 1, 1952, pp. 679-685. MR 0044468 (13:422a)
  • [13] -, Introduction to metamathematics, New York and Toronto, Van Nostrand, Amsterdam, North-Holland, and Groningen, Noordhoff, 1952, X+550 pp. MR 0051790 (14:525m)
  • [14] -, Arithmetical predicates and function quantifiers, Trans. Amer. Math. Soc. vol. 79 (1955) pp. 312-340. For errata, cf. ibid. p. 386 and vol. 81 (1956) p. 524, and Proc. Amer. Math. Soc. vol. 8 (1957) p. 1006. MR 0070594 (17:4g)
  • [15] -, On the forms of the predicates in the theory of constructive ordinals (second paper), Amer. J. Math. vol. 77 (1955) pp. 405-428. Errata: p. 408 1. 7 for `` $ {\text{y}}$'' read ``$ y$", p. 412 1. 1 for `` $ \operatorname{Im} (k)$'' read `` $ \overline {\operatorname{Im} } (k)$", p. 425 for second ``$ \beta $'' 1. 15 and ``$ \beta $'' 1. 16 read `` $ \bar \beta $"; also ``$ \cong $'' should be ``$ \simeq $'' except on p. 411 1. 9 (namely on pp. 408, 411 1. 11, 412-414, 418, 427). MR 0070595 (17:5a)
  • [16] -, Hierarchies of number-theoretic predicates, Bull. Amer. Math. Soc. vol. 61 (1955) pp. 193-213. MR 0070593 (17:4f)
  • [17] -, Extension of an effectively generated class of functions by enumeration, Colloquium Mathematicum (Wrocław), vol. 6.
  • [18] -, Countable functionals, Constructivity in Mathematics, Amsterdam, North-Holland.
  • [19] S. C. Kleene and E. L. Post, The upper semi-lattice of degrees of recursive unsolvability, Ann. of Math. (2) vol. 59 (1954) pp. 379-407. Erratum: p. 404, the next to the last ``$ = $'' in (56) should be ``$ \ne $". MR 0061078 (15:772a)
  • [20] D. König, Über eine Schlussweise aus dem Endlichen ins Unendliche, Acta Sci. Math. Szeged vol. 3 (1927) pp. 121-130.
  • [21] G. Kreisel, Constructive interpretations of analysis by means of functionals of finite type, Constructivity in Mathematics, Amsterdam, North-Holland.
  • [22] R. Péter, Über den Zusammenhang der verschiedenen Begriffe der rekursiven Funktion, Math. Ann. vol. 110 (1934) pp. 612-632.
  • [23] -, Rekursive Funktionen, Budapest, Akademischer Verlag, 1951, 206 pp. MR 0044467 (13:421l)
  • [24] -, Probleme der Hilbertschen Theorie der höheren Stufen von rekursiven Funktionen, Acta Math. Acad. Sci. Hungar. vol. 2 (1951) pp. 247-274. MR 0053042 (14:713d)
  • [25] E. L. Post, Degrees of recursive unsolvability (Preliminary report), Bull. Amer. Math. Soc. Abstract 54-7-269.
  • [26] C. Spector, Recursive well-orderings, J. Symb. Logic, vol. 20 (1955) pp. 151-163. MR 0074347 (17:570b)
  • [27] -, On degrees of recursive unsolvability, Ann. of Math. (2) vol. 64 (1956) pp. 581-592. MR 0082457 (18:552d)
  • [28] A. Tarski, Einige Betrachtungen über die Begriffe der $ \omega $-Widerspruchsfreiheit und der $ \omega $-Vollständigkeit, Monatshefte für Mathematik und Physik vol. 40 (1933) pp. 97-112. MR 1550192
  • [29] -, A problem concerning the notion of definability, J. Symb. Logic vol. 13 (1948) pp. 107-111. MR 0026625 (10:176b)
  • [30] A. M. Turing, Systems of logic based on ordinals, Proc. London Math. Soc. (2) vol. 45 (1939) pp. 161-228.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 02.00

Retrieve articles in all journals with MSC: 02.00


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1959-0102480-9
PII: S 0002-9947(1959)0102480-9
Article copyright: © Copyright 1959 American Mathematical Society