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Recursive well-orderings

Published online by Cambridge University Press:  12 March 2014

Clifford Spector
Clifford Spector*
Affiliation:
The University of Wisconsin and The Ohio State University
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Extract

Cantor's second ordinal number class is perhaps the simplest example of a set of mathematical objects which cannot all be named in one language. In this paper we shall investigate a system of names for a segment of the first and second number classes in relation to decision problems. The system, except for one minor difference, is the one studied by Markwald in [12]. In our system ordinals are named by natural numbers from a set W via recursive well-orderings of subsets of the natural numbers.

The decision problems will be related to the hyperarithmetical hierarchy of Davis [2], [3] and Kleene [8]. This hierarchy is indexed by ordinal notations from Kleene's system S3 [4], [6], [9], in which ordinals are named by natural numbers from a set O, partially well-ordered ([12] p. 138) by a relation a≤Ob; O and ≤O are defined inductively by applications of the successor and limit operations. As results of this investigation, we shall (i) answer negatively Markwald's question [12] Theorem 12 whether his set “W” is arithmetical by showing that it is not even hyperarithmetical, (ii) obtain a new proof of the main result of Kleene [10] that every predicate expressible in both the one-function-quantifier forms of [8] is recursive in Hα for some aεO, (iii) answer affirmatively the question raised by Davis [2], [3] whether all the Church-Kleene constructive ordinals are uniqueness ordinals, and (iv) solve the function-quantifier analog of Post's problem [15]. Strong use will be made of the well-orderings that can be constructed from one-function-quantifier predicates as in [9].

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 20 , Issue 2 , June 1955 , pp. 151 - 163
Copyright
Copyright © Association for Symbolic Logic 1955

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References

BIBLIOGRAPHY

[1]Birkhoff, Garrett, Lattice theory, American Mathematical Society Colloquium Series, vol. 25, revised ed., 1948, xiv + 283 pp.Google Scholar
[2]Davis, Martin, On the theory of recursive unsolvability, Ph. D. thesis, Princeton University, 1950.Google Scholar
[3]Davis, Martin, Relatively recursive junctions and the extended Kleene hierarchy, Proceedings of the International Congress of Mathematicians (Cambridge, Mass., U.S.A., Aug. 30-Sept. 6, 1950), 1952, vol. 1, p. 723.Google Scholar
[4]Kleene, S. C., On notation for ordinal numbers, this Journal, vol. 3 (1938), pp. 150–155.Google Scholar
[5]Kleene, S. C., Recursive predicates and quantifiers, Transactions of the American Mathematical Society, vol. 53 (1943), pp. 41–73. (Omit § 15 because of the error in [6].)Google Scholar
[6]Kleene, S. C., On the forms of the predicates in the theory of constructive ordinals, American journal of mathematics, vol. 66 (1944), pp. 41–58. (For an error affecting §§ 8, 14–16, cf. [7] p. 527 or [9].)CrossRefGoogle Scholar
[7]Kleene, S. C., Introduction to metamathematics, New York and Toronto (Van Nostrand), Amsterdam (North-Holland Publishing Company), and Groningen (Noordhoff), 1952, x + 550 pp.Google Scholar
[8]Kleene, S. C., Arithmetical predicates and function quantifiers, to appear in Transactions of the American Mathematical Society.Google Scholar
[9]Kleene, S. C., On the forms of the predicates in the theory of constructive ordinals (second paper), to appear in American journal of mathematics.Google Scholar
[10]Kleene, S. C., Hierarchies of number-theoretic predicates, to appear in Bulletin of the American Mathematical Society.Google Scholar
[11]Kleene, S. C. and Post, Emil L., The upper semi-lattice of degrees of recursive unsolvability, Annals of mathematics, vol. 59 (1954), pp. 379–407.Google Scholar
[12]Markwald, Werner, Zur Theorie der konstruktiven Wohlordnungen, Mathematische Annalen, vol. 127 (1954), pp. 135–149.Google Scholar
[13]Mostowski, Andrzej, On definable sets of positive integers, Fundamenta mathematicae, vol. 34 (1947), pp. 81–112.CrossRefGoogle Scholar
[14]Mostowski, Andrzej, A classification of logical systems, Studia philosophica, vol. 4 (1951), pp. 237–274.Google Scholar
[15]Post, Emil L., Recursively enumerable sets of positive integers and their decision problems, Bulletin of the American Mathematical Society, vol. 50 (1944), pp. 284–316.Google Scholar