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A noninitial segment of index sets

Published online by Cambridge University Press:  12 March 2014

Louise Hay
Louise Hay*
Affiliation:
University of Illinois at Chicago Circle, Chicago, Illinois 60680
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Extract

Let {Wk}k ≥ 0 be a standard enumeration of all recursively enumerable (r.e.) sets. If A is any class of r.e. sets, let θA denote the index set of A, i.e., θA = {k ∣ Wk ∈ A}. The one-one degrees of index sets form a partial order ℐ which is a proper subordering of the partial order of all one-one degrees. Denote by ⌀ the one-one degree of the empty set, and, if b is the one-one degree of θB, denote by the one-one degree of . Let . Let {Ym}m≥0 be the sequence of index sets of nonempty finite classes of finite sets (classified in [5] and independently, in [2]) and denote by am the one-one degree of Ym. As shown in [2], these degrees are complete at each level of the difference hierarchy generated by the r.e. sets. It was proved in [3] that, for each m ≥ 0,

(a) am+1 and ām+1 are incomparable immediate successors of am and ām, and

(b) .

For m = 0, since Y0 = θ{⌀}, it follows from (a) that

(c) .

Hence it follows that

(d) {⌀, , ao, ā0, a1, ā1 is an initial segment of ℐ.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 39 , Issue 2 , June 1974 , pp. 209 - 224
Copyright
Copyright © Association for Symbolic Logic 1974

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References

REFERENCES

[1] Dekker, J. C. E. and Myhill, J., Some theorems on classes of recursively enumerable sets, Transactions of the American Mathematical Society, vol. 89 (1958), pp. 25–59.Google Scholar
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[3] Hay, L., A discrete chain of degrees of index sets, this Journal, vol. 37 (1972), pp. 139–149.Google Scholar
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[5] Hay, L., Index sets of finite classes of recursively enumerable sets, this Journal, vol. 34 (1967), pp. 39–44.Google Scholar
[6] Rogers, H. Jr., Theory of recursive functions and effective computability, McGraw-Hill, New York, 1967.Google Scholar