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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References | Similar Articles | Additional Information

**[10]**S. C. Kleene,*Recursive predicates and quantifiers*, Trans. Amer. Math. Soc.**53**(1943), 41–73. MR**7371**, https://doi.org/10.1090/S0002-9947-1943-0007371-8**[IM]**Stephen Cole Kleene,*Introduction to metamathematics*, D. Van Nostrand Co., Inc., New York, N. Y., 1952. MR**0051790****[14]**S. C. Kleene,*Arithmetical predicates and function quantifiers*, Trans. Amer. Math. Soc.**79**(1955), 312–340. MR**70594**, https://doi.org/10.1090/S0002-9947-1955-0070594-4**[16]**S. C. Kleene,*Hierarchies of number-theoretic predicates*, Bull. Amer. Math. Soc.**61**(1955), 193–213. MR**70593**, https://doi.org/10.1090/S0002-9904-1955-09896-3**[17]**S. C. Kleene,*Extension of an effectively generated class of functions by enumeration*, Colloq. Math.**6**(1958), 67–78. MR**118672**, https://doi.org/10.4064/cm-6-1-67-78**[18]**S. C. Kleene,*Countable functionals*, Constructivity in mathematics: Proceedings of the colloquium held at Amsterdam, 1957 (edited by A. Heyting), Studies in Logic and the Foundations of Mathematics, North-Holland Publishing Co., Amsterdam, 1959, pp. 81–100. MR**0112837****[19]**S. C. Kleene and Emil L. Post,*The upper semi-lattice of degrees of recursive unsolvability*, Ann. of Math. (2)**59**(1954), 379–407. MR**61078**, https://doi.org/10.2307/1969708**[25]**E. L. Post,*Degrees of recursive unsolvability*, Abstract 54-7-269, Bull. Amer. Math. Soc.**54**(1948), 641-642.**[26]**Clifford Spector,*Recursive well-orderings*, J. Symbolic Logic**20**(1955), 151–163. MR**74347**, https://doi.org/10.2307/2266902**[27]**Clifford Spector,*On degrees of recursive unsolvability*, Ann. of Math. (2)**64**(1956), 581–592. MR**82457**, https://doi.org/10.2307/1969604**[31]**J. W. Addison,*Separation principles in the hierarchies of classical and effective descriptive set theory*, Fund. Math.**46**(1959), 123–135. MR**131357**, https://doi.org/10.4064/fm-46-2-123-135**[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**91243**, https://doi.org/10.1090/S0002-9939-1957-0091243-2**[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]**S. C. Kleene,*Recursive functionals and quantifiers of finite types. I*, Trans. Amer. Math. Soc.**91**(1959), 1–52. MR**102480**, https://doi.org/10.1090/S0002-9947-1959-0102480-9**[38]**S. C. Kleene,*Mathematical logic: constructive and non-constructive operations*, Proc. Internat. Congress Math., Cambridge Univ. Press, New York, 1960, pp. 137–153. MR**0114750****[39]**S. C. Kleene,*Turing-machine computable functionals of finite types. I*, Logic, Methodology and Philosophy of Science (Proc. 1960 Internat. Congr .), Stanford Univ. Press, Stanford, Calif., 1962, pp. 38–45. MR**0186542****[40]**S. C. Kleene,*Turing-machine computable functionals of finite types. II*, Proc. London Math. Soc. (3)**12**(1962), 245–258. MR**0186543**, https://doi.org/10.1112/plms/s3-12.1.245**[41]**S. C. Kleene,*Lambda-definable functionals of finite types*, Fund. Math.**50**(1961/62), 281–303. MR**186544**, https://doi.org/10.4064/fm-50-3-281-303**[42]**S. C. Kleene,*Herbrand-Gödel-style recursive functionals of finite types*, Proc. Sympos. Pure Math., Vol. V, American Mathematical Society, Providence, R.I., 1962, pp. 49–75. MR**0141594****[43]**Donald L. Kreider and Hartley Rogers Jr.,*Constructive versions of ordinal number classes*, Trans. Amer. Math. Soc.**100**(1961), 325–369. MR**151396**, https://doi.org/10.1090/S0002-9947-1961-0151396-X**[44]**J. R. Shoenfield,*The form of the negation of a predicate*, Proc. Sympos. Pure Math., Vol. V, American Mathematical Society, Providence, R.I., 1962, pp. 131–134. MR**0142449****[45]**Tosiyuki Tugué,*Predicates recursive in a type-2 object and Kleene hierarchies*, Comment. Math. Univ. St. Paul.**8**(1960), 97–117. MR**110639****[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