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Recursively enumerable m- and tt-degrees. I: The quantity of m-degrees

Published online by Cambridge University Press:  12 March 2014

R. G. Downey
R. G. Downey*
Affiliation:
Department of Mathematics, Victoria University, Wellington, New Zealand
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Extract

In [1], Degtëv constructed a nonzero r.e. tt-degree containing a single r.e. m-degree. It is not difficult to construct an r.e. tt-degree containing infinitely many r.e. m-degrees (Fischer [6]); indeed, in [3], the author constructed an r.e. tt-degree with no greatest r.e. m-degree. Odifreddi [12, Problem 10] asked if every r.e. tt-degree contains either one or infinitely many r.e. m-degrees. The goal of this paper is to solve Odifreddi's question by showing:

Theorem. There exists a nonzero r.e. tt-degree containing exactly 3 r.e. m-degrees.

This theorem can be extended to show that there exist r.e. tt-degrees with arbitrarily large finite numbers of r.e. m-degrees.

We remark that save for the aforementioned results, very little is known about the structures that can be realized as the collection of r.e. m-degrees within an r.e. tt-degree. It seems conceivable that the methods of the present paper may be useful in, for example, embedding distributive (semi) lattices into such structures.

In part II of this paper [4], we continue our analysis of r.e. m- and tt-degrees. We define an r.e. tt-degree to be singular if it contains a single r.e. m-degree, and an r.e. T-degree a to be singular if a contains a singular r.e. tt-degree.

In [4] we study the distribution (in the r.e. T-degrees) of singular tt-degrees. We show that 0′T is singular (solving a question of Odifreddi [11]), and that the singular T-degrees are dense, but also we construct a nonsingular T-degree. The techniques used for the first results extend those of §2 of the present paper.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 54 , Issue 2 , June 1989 , pp. 553 - 567
Copyright
Copyright © Association for Symbolic Logic 1989

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References

REFERENCES

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