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A metatheorem for constructions by finitely many workers

Published online by Cambridge University Press:  12 March 2014

J. F. Knight
J. F. Knight*
Affiliation:
Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556
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Extract

The aim of the present paper is to give some general conditions for constructions by finitely many workers. Constructions using infinitely many workers will not be considered here, although there are examples of such constructions. The original construction using the method of workers, due to Harrington [H], has a worker n for each n ∈ ω, as do the constructions in [K1] and [K2]. Marker [M] obtains a result using three workers. In [AJK], there are two constructions that use three workers. There are also two constructions that have, for an arbitrary recursive ordinal α, one worker for each β < α.

The main result here is a metatheorem, which is patterned after Proposition 1 of Ash [A]. As in [A], the object of the construction is to attach “labels” to the nodes in a highly nonrecursive path through a tree, while recursively enumerating neighborhoods of an “adherent” point in a metric space. There is a family of relations associated with the labels, and the metatheorem here and the one in [A] both say that the construction will succeed if these relations satisfy a list of properties. There are significant differences between the result here and that in [A]. One difference is that certain relations which in [A] were required to be r.e. need not be r.e. here. Another difference is that there are extra relations here, and as a result, the list of properties to be satisfied is longer and more horrible than that in [A].

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 55 , Issue 2 , June 1990 , pp. 787 - 804
Copyright
Copyright © Association for Symbolic Logic 1990

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References

REFERENCES

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