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Π01-classes and Rado's selection principle

Published online by Cambridge University Press:  12 March 2014

C. G. Jockusch ,
A. Lewis  and
J. B. Remmel
C. G. Jockusch
Affiliation:
Department of Mathematics, University of Illinois, Urbana, Illinois 61801
A. Lewis
Affiliation:
Department of Mathematics, Cornell University, Ithaca, New York 14853 Department of Mathematical Social Sciences, University of California at Irvine, Irvine, California 92717
J. B. Remmel
Affiliation:
Department of Mathematics, University of California at San Diego, La Jolla, California 92037 Department of Mathematics, University of Florida, Gainesville, Florida 32611
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Extract

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<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.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 56 , Issue 2 , June 1991 , pp. 684 - 693
Copyright
Copyright © Association for Symbolic Logic 1991

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