Quantification of number-theoretic functions
Compositio Mathematica, Tome 14 (1959-1960), pp. 23-40. 
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Kleene, S. C. Quantification of number-theoretic functions. Compositio Mathematica, Tome 14 (1959-1960), pp. 23-40. http://www.numdam.org/item/CM_1959-1960__14__23_0/

J.W. Addison and S.C. Kleene [1] A note on function quantification, Proc. Amer. Math. Soc., vol. 8 (1957), pp. 1002-1006. | Zbl

A. Church [2] Introduction to mathematical logic, vol. 1, Princeton, N. J. (Princeton Univ. Press), 1956, x + 376 pp. | MR | Zbl

A. Church and S.C. Kleene [3] Formal definitions in the theory of ordinal numbers, Fund. Math., vol. 28 (1936), pp. 11-21. In the definition of ×O on p. 18, the case b is 0O should have been treated separately (so a X O 0O conv 0O). | JFM | Zbl

S.C. Kleene [4] Introduction to metamathematics, Amsterdam (North Holland), Groningen (Noordhoff), New York and Toronto (Van Nostrand), 1952, X + 550 pp. | MR | Zbl

[5] Arithmetical predicates and function quantifiers, Trans. Amer. Math. Soc., vol. 79 (1955), pp. 312-340.Errata, ibid., vol. 80 (1955), p. 386 and vol. 81 (1956), p. 524, and in Reference 5 of [1], p. 1006. | MR | Zbl

[6] On the forms of the predicates in the theory of constructive ordinals (second paper), vol. 77 (1955), pp. 405-428. | Zbl

[7] Hierarchies of number-theoretic predicates, Bull. Amer. Math. Soc., vol. 61 (1955), pp. 193-213. | Zbl

S.C. Kleene and E.L. Post [8] The upper semi-lattice of degrees of recursive unsolvability, Ann. of Math. (2), vol. 59 (1954), pp. 379-407. | Zbl

C. Spector [9] Recursive well-orderings, Jour. Symbolic Logic, vol. 20 (1955), pp. 151-163. | MR | Zbl