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On analytic well-orderings

Published online by Cambridge University Press:  12 March 2014

Hisao Tanaka
Hisao Tanaka*
Affiliation:
Hosei University, Koganei, Tokyo, Japan
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Extract

Since about 1955, recursive, (hyper-)arithmetic and well-orderings have been investigated by many authors. Now we shall generally consider analytic well-orderings and compare their representation-capacities for ordinals.

Let K be a class of predicates. Then, R is called a Kwell-ordering if R is a binary relation of natural numbers belonging to the class K and satisfies the following conditions:

(i) R(x, y) ∧ R(y, x) → x = y;

(ii) x, y ∈ D(R) → R(x, y) ∨ R(y, x), where D(R) is the domain of R, that is to say, the set {x ∣ (∃y)[R(x, y) ∨ R(y, x)]};

(iii) R(x, y) ∧ R(y, z) → R(x, z);

(iv) .

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 35 , Issue 2 , June 1970 , pp. 198 - 204
Copyright
Copyright © Association for Symbolic Logic 1970

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References

[1]Addison, J. W., Some consequences of the axiom of constructibility, Fundamenta mathematicae, vol. 46 (1959), pp. 337–357.CrossRefGoogle Scholar
[2]Addison, J. W. and Kleene, S. C., A note on function quantification, Proceedings of the American Mathematical Society, vol. 8 (1957), pp. 1002–1006.CrossRefGoogle Scholar
[3]Feferman, S. and Spector, C., Incompleteness along paths in progressions of theories, this Journal, vol. 27 (1962), pp. 383–390.Google Scholar
[4]Gandy, R. O., Proof of Mostowsk's conjecture, Bulletin de l' Académie Polonaise des Science. Série des Sciences Mathématiques, Astronomique et Physiques, vol. 8 (1960), pp. 571–575.Google Scholar
[5]Kleene, S. C., Introduction to metamathematics, North-Holland, Amsterdam, 1952.Google Scholar
[6]Kleene, S. C., Arithmetical predicates and function quantifiers, Transactions of the American Mathematical Society, vol. 79 (1955), pp. 312–340.CrossRefGoogle Scholar
[7]Kleene, S. C., On the forms of the predicates in the theory of constructive ordinals (second paper), American journal of mathematics, vol. 77 (1955), pp. 405–428.CrossRefGoogle Scholar
[8]Kuratowski, C., Topologie. I, 4th ed., PWN, Warsaw, 1958.Google Scholar
[9]Richter, W., Extensions of the constructive ordinals, this Journal, vol. 30 (1965), pp. 193–211.Google Scholar
[10]Rogers, H. Jr., Theory of recursive functions and effective computability, McGraw-Hill, New York, 1967.Google Scholar
[11]Shoenfield, J. R., Mathematical logic, Addison-Wesley, Reading, Mass., 1967.Google Scholar
[12]Spector, C., Recursive well-orderings, this Journal, vol. 20 (1955), pp. 151–163.Google Scholar
[13]Suzuki, Y., A complete classification of the -functions, Bulletin of the American Mathematical Society, vol. 70 (1964), pp. 246–253.CrossRefGoogle Scholar