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Π01-classes and Rado's selection principle

  • C. G. Jockusch (a1), A. Lewis (a2) (a3) and J. B. Remmel (a4) (a5)

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There are several areas in recursive algebra and combinatorics in which bounded or recursively bounded -classes have arisen. For our purposes we may define a -class to be a set Path(T) of all infinite paths through a recursive tree T. Here a recursive tree T is just a recursive subset of ω, the set of all finite sequences of the natural numbers ω = {0,1,2,…}, which is closed under initial segments. If the tree T is finitely branching, then we say the -class Path(T) is bounded. If T is highly recursive, i.e., if there exists a partial recursive function f: T→ω such that for each node ηЄ T, f(η) equals the number of immediate successors of η, then we say that the -class Path(T) is recursively bounded (r.b.). For example, Manaster and Rosenstein in [6] studied the effective version of the marriage problem and showed that the set of proper marriages for a recursive society S was always a bounded -class and the set of proper marriages for a highly recursive society was always an r.b. -class. Indeed, Manaster and Rosenstein showed that, in the case of the symmetric marriage problem, any r.b. -class could be represented as the set of symmetric marriages of a highly recursive society S in the sense that given any r.b. Π1-class C there is a society Sc such that there is a natural, effective, degree-preserving 1:1 correspondence between the elements of C and the symmetric marriages of Sc. Jockusch conjectured that the set of marriages of a recursive society can represent any bounded -class and the set of marriages of a highly recursive society can represent any r.b. -class. These conjectures remain open. However, Metakides and Nerode [7] showed that any r.b. -class could be represented by the set of total orderings of a recursive real field and vice versa that the set of total orderings of a recursive real field is always an r.b. -class.

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[1]Bean, D., Effective coloration, this Journal, vol. 41 (1976), pp. 469480.
[2]Jockusch, C. G. and McLaughlin, T. G., Countable retracing functions and predicates, Pacific Journal of Mathematics, vol. 30 (1969), pp. 6793.
[3]Jockusch, C. G. and Soare, R. I., -classes and degrees of theories, Transactions of the American Mathematical Society, vol. 173 (1972), pp. 3356.
[4]Jockusch, C. G. and Soare, R. I., Degrees of members of -classes, Pacific Journal of Mathematics, vol. 40 (1972), pp. 605616.
[5]Kučera, A., An alternative, priority-free, solution to Post's problem, Mathematical foundations of computer science (twelfth symposium, Bratislava, 1986), edited by Gruska, J.et al., Lecture Notes in Computer Science, vol. 233, Springer-Verlag, Berlin, 1986, pp. 493500.
[6]Manaster, A. and Rosenstein, J., Effective matchmaking, Proceedings of the London Mathematical Society, ser. 3 vol. 25 (1972), pp. 615654.
[7]Metakides, G. and Nerode, A., Effective content of field theory, Annals of Mathematical Logic, vol. 11 (1979), 147171.
[8]Milner, E. C., Selectivity and weakly compact cardinals, Bulletin of the London Mathematical Society, vol. 14 (1982), pp. 329333.
[9]Rado, R., Axiomatic treatment of rank in infinite sets, Canadian Journal of Mathematics, vol. 1 (1949), pp. 337343.
[10]Rado, R., A selection lemma, Journal of Combinatorial Theory, vol. 10 (1971), pp. 176177.
[11]Rado, R., Selective families of sets, Proceedings of the Royal Society of London, Series A, vol. 372 (1980), pp. 307315.
[12]Rav, Y., Variants of Rado's selection lemma and their applications, Mathematische Nachrichten, vol. 79 (1977), pp. 145165.
[13]Remmel, J. B., Graph colorings and recursively bounded -classes, Annals of Pure and Applied Logic, vol. 32 (1986), pp. 185194.
[14]Rogers, H., Theory of recursive functions and effective computability, McGraw-Hill, New York, 1967.
[15]Shoenfield, J. R., Degrees of models, this Journal, vol. 25 (1960), pp. 233237.
[16]Yates, C. E. M., Arithmetical sets and retracing functions, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 13 (1967), pp. 193204.

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Π01-classes and Rado's selection principle

  • C. G. Jockusch (a1), A. Lewis (a2) (a3) and J. B. Remmel (a4) (a5)

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