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Countable initial segments of the degrees of unsolvability

Published online by Cambridge University Press:  12 March 2014

A. H. Lachlan  and
R. Lebeuf
A. H. Lachlan
Affiliation:
Simon Fraser University, Burnaby 2, British Columbia, Canada
R. Lebeuf
Affiliation:
College Marie-Victorin, 7000 Marie-Victorin Montreal, Quebec, Canada
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Extract

In this paper we show that any countable upper semilattice with zero can be embedded as an initial segment of the degrees of unsolvability. This provides a characterization of the order types of the countable initial segments of the degrees since any such initial segment is trivially an initial segment of a countable upper semilattice. Let a segment S of the degrees be a set of degrees such that, if a, b Є S and a < c < b, then c Є S. One may readily observe that our result also characterizes the order types of all countable segments of the degrees of unsolvability.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 41 , Issue 2 , June 1976 , pp. 289 - 300
Copyright
Copyright © Association for Symbolic Logic 1976

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References

REFERENCES

[1] Lerman, M., Initial segment of the degrees of unsolvability, Annals of Mathematics, vol. 93 (1971), pp. 365–389.CrossRefGoogle Scholar
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[3] Lachlan, A. H., Recursion theory seminar (Warsaw, 1973), mimeographed notes.Google Scholar
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