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# On analytic well-orderings

## 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, yD(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) .

## References

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[1]Addison, J. W., Some consequences of the axiom of constructibility, Fundamenta mathematicae, vol. 46 (1959), pp. 337357.
[2]Addison, J. W. and Kleene, S. C., A note on function quantification, Proceedings of the American Mathematical Society, vol. 8 (1957), pp. 10021006.
[3]Feferman, S. and Spector, C., Incompleteness along paths in progressions of theories, this Journal, vol. 27 (1962), pp. 383390.
[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. 571575.
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[6]Kleene, S. C., Arithmetical predicates and function quantifiers, Transactions of the American Mathematical Society, vol. 79 (1955), pp. 312340.
[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. 405428.
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[9]Richter, W., Extensions of the constructive ordinals, this Journal, vol. 30 (1965), pp. 193211.
[10]Rogers, H. Jr., Theory of recursive functions and effective computability, McGraw-Hill, New York, 1967.
[12]Spector, C., Recursive well-orderings, this Journal, vol. 20 (1955), pp. 151163.
[13]Suzuki, Y., A complete classification of the -functions, Bulletin of the American Mathematical Society, vol. 70 (1964), pp. 246253.

# On analytic well-orderings

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