Recursive functionals and quantifiers of finite types. II

Author:
S. C. Kleene

Journal:
Trans. Amer. Math. Soc. **108** (1963), 106-142

MSC:
Primary 02.70

DOI:
https://doi.org/10.1090/S0002-9947-1963-0153557-4

MathSciNet review:
0153557

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**[10]**S. C. Kleene,*Recursive predicates and quantifiers*, Trans. Amer. Math. Soc.**53**(1943), 41-73. MR**0007371 (4:126b)****[IM]**-,*Introduction to metamathematics*, Van Nostrand, New York, North-Holland, Amsterdam, Noordhoff, Groningen, 1952. MR**0051790 (14:525m)****[14]**-,*Arithmetical predicates and function quantifiers*, Trans. Amer. Math. Soc.**79**(1955), 312-340. MR**0070594 (17:4g)****[16]**-,*Hierarchies of number-theoretic predicates*, Bull. Amer. Math. Soc.**61**(1955), 193-213. MR**0070593 (17:4f)****[17]**-,*Extension of an effectively generated class of functions by enumeration*, Colloq. Math.**6**(1958), 67-78. Errata: P. 68 l. 17, for `` "read" "; the continuation from p. 67 of Footnote 1 should begin ``(Added 17 December 1957.)", and in the fifth line the final ``'' should be ``". P. 69 l. 3 from bottom of text, for ``'' read ``". P. 70 l. 4, for `` '' read `` "; first line of §4, delete ``a"; fourth line of Footnote 6, for ``'' read ``". P. 74 in (15), for ``'' read ``". P. 75 l. 8, for ``'' read ``"; l. 5 from below, for ``'' read `` ". P. 76 l. 11, for ``'' read ``"; l. 12, for ``'' read ``"; in the definition of , for `` '' read `` "; l. 17 from below, for ``1'*s*'' read ``1's". P. 77 top of display, for ``*l*'' read ``1"; third line of P 237, the inferior ``*O*'' should be inferior to inferior, and for ``*y, s*'' read ``*y's*". P. 78 l. 1, the inferior ``*O*'' should be inferior to inferior. MR**0118672 (22:9443)****[18]**-,*Countable functionals*, Constructivity in Mathematics, pp. 81-100, North-Holland, Amsterdam, 1959. Errata: P. 81, in the third line of the publishers' note, for ``281, 285'' read ``285, 281"; third line of Footnote 2, for `` '' read `` "; p. 83 definition of middle line, for first ``'' read `` "; p. 86 proof of theorem third line, for `` '' read `` "; p. 94 in both formulas displayed in Case 6, change ``2'' to ``", the inferior-to-superior ``*k*'' to inferior-to-superior ``0", the inferior ``'' to inferior ``*i*", and the next (inferior-to-superior) ``*i*'' to ``", and in the second line of the first of these formulas, the superior ``*i*'' should be inferior to superior. MR**0112837 (22:3686)****[19]**S. C. Kleene and E. L. Post,*The upper semi-lattice of degrees of recursive unsolvability*, Ann. of Math. (2)**59**(1954), 379-407. MR**0061078 (15:772a)****[25]**E. L. Post,*Degrees of recursive unsolvability*, Abstract 54-7-269, Bull. Amer. Math. Soc.**54**(1948), 641-642.**[26]**C. Spector,*Recursive well-orderings*, J. Symbolic Logic**20**(1955), 151-163. MR**0074347 (17:570b)****[27]**-,*On degrees of recursive unsolvability*, Ann. of Math. (2)**64**(1956), 581-592. MR**0082457 (18:552d)****[31]**J. W. Addison,*Separation principles in the hierarchies of classical and effective descriptive set theory*, Fund. Math.**46**(1958), 123-135. MR**0131357 (24:A1209)****[32]**-,*On the Novikov-Kondo theorem*, abstract of a paper delivered at the Symposium ``Infinitistic methods", Warsaw, 2-9 September 1959, 2 pp. (mimeographed).**[33]**J. W. Addison and S. C. Kleene,*A note on function quantification*, Proc. Amer. Math. Soc.**8**(1957), 1002-1006. MR**0091243 (19:934b)****[34]**A. Church and S. C. Kleene,*Formal definitions in the theory of ordinal numbers*, Fund. Math.**28**(1936), 11-21.**[35]**D. A. Clarke,*Hierarchies of predicates of arbitrary finite types*, Univ. of Wisconsin, Ph. D. thesis (in preparation).**[36]**S. C. Kleene,*On notation for ordinal numbers*, J. Symbolic Logic**3**(1938), 150-155.**[37]**-,*Recursive functional and quantifiers of finite types*I, Trans. Amer. Math. Soc.**91**(1959), 1-52. Errata: P. 7 l. 16, for ``on'' read ``in"; l. 2 from bottom of text, for `` '' read `` ". P. 9 first line of 2.4, for `` '' read `` ". P. 16 just above third display, for ``are false'' read ``"; just below, for ``index'' read ``an index". P. 18 last two lines of XXI, for ``*with*'' read `` ". P. 20 l. 5, for ``'' read ``"; l. 6, change first ``'' to ``". P. 21 l. 4 from below, for first `` '' read `` ". P. 23 l. 9, for ``)'' read ``]". On p. 24 in the right column of the table for Case 6, and on p. 25 in the display l. 8 from below, change ``2'' to ``", the inferior-to-superior ``*k*'' to inferior-to-superior ``0", the inferior ``'' to inferior ``*i*", and the next (inferior-to-superior) ``*i*'' to ``". P. 25 l. 2, the superior colon should be a superior semicolon. P. 30 l. 6 from below, for ``second'' read ``last". P. 37 l. 10 from below, for ``'' read ``". P. 43 l. 1, for `` '' read `` ". P. 46, read ``'' instead of ``='' in lines 4, 10, 12 and in the two left occurences in (76), P. 51 Item 14, for ``ibid.'' read ``ibid. vol. 80". P. 52 Item 21, the title should read ``*Interpretation of analysis by means of constructive functionals of finite types*". MR**0102480 (21:1273)****[38]**-,*Mathematical logic*:*constructive and non-constructive operations*, Proceedings of the International Congress of Mathematicians, 14-21 August 1958, Cambridge, Cambridge Univ. Press, 1960, pp. 137-153. Errata: P. 146 l. 1, for ``*proavble*'' read ``*provable*"; p. 149 l. 8, for `` '' read `` ". MR**0114750 (22:5569)****[39]**-,*Turing-machine computable functionals of finite types*I, Logic, Methodology and Philosophy of Science, Proceedings of the 1960 International Congress, Stanford, Calif., August 24-September 2, Stanford Univ. Press, Stanford, Calif., 1962, pp. 38-45. MR**0186542 (32:4001)****[40]**-,*Turing-machine computable functionals of finite types*II, Proc. London Math. Soc. (3)**12**(1962), 245-258. MR**0186543 (32:4002)****[41]**-,*Lambda-definable functionals of finite types*, Fund. Math.**50**(1962), 281-303. MR**0186544 (32:4003)****[42]**-,*Herbrand-Gödel-style recursive functionals of finite types*, Recursive function theory, Proc. Sympos. Pure Math. Vol. 5, pp. 49-75, Amer. Math. Soc., Providence, R. I., 1962. MR**0141594 (25:4992)****[43]**D. L. Kreider and Hartley Rogers, Jr.,*Constructive versions of ordinal number classes*, Trans. Amer. Math. Soc.**100**(1961), 325-369. MR**0151396 (27:1381)****[44]**J. R. Shoenfield,*The form of the negation of a predicate*, Recursive function theory, Proc. Sympos. Pure Math. Vol. 5, pp. 131-134, Amer. Math. Soc., Providence, R. I., 1963. MR**0142449 (26:18)****[45]**Tosiyuki Tugué,*Predicates recursive in a type*-2*object and Kleene hierarchies*, Comment. Math. Univ. St. Paul. (Tokyo)**8**(1960), 97-117. MR**0110639 (22:1515)****[46]**Hao Wang,*Remarks on constructive ordinals and set theory*, Summaries of Talks Presented at the Summer Institute of Symbolic Logic in 1957 at Cornell University (mimeographed 1957; 2nd ed., Communications Research Division, Institute for Defense Analyses, 1960) pp. 383-390.

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DOI:
https://doi.org/10.1090/S0002-9947-1963-0153557-4

Article copyright:
© Copyright 1963
American Mathematical Society