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An extension of the nondiamond theorem in classical and α-recursion theory

Published online by Cambridge University Press:  12 March 2014

Klaus Ambos-Spies
Klaus Ambos-Spies*
Affiliation:
Universität Dortmund, Dortmund, West Germany
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Extract

Lachlan's nondiamond theorem [7, Theorem 5] asserts that there is no embedding of the four-element Boolean algebra (diamond) in the recursively enumerable degrees which preserves infima, suprema, and least and greatest elements. Lachlan observed that, essentially by relativization, the theorem can be extended to

Using the Sacks splitting theorem he concluded that there exists a pair of r.e. degrees which does not have an infimum, thus showing that the r.e. degrees do not form a lattice.

We will first prove the following extension of (1):

where an r.e. degree a is non-b-cappable if . From (2) we obtain more information about pairs of r.e. degrees without infima: For every nonzero low r.e. degree there exists an incomparable one such that the two degrees do not have an infimum and there is an r.e. degree which is not half of a pair of incomparable r.e. degrees which has an infimum in the low r.e. degrees. Probably the most interesting corollary of (2) is that the join of any cappable r.e. degree (i.e. half of a minimal pair) and any low r.e. degree is incomplete. Consequently there is an incomplete noncappable degree above every incomplete r.e. degree. Cooper's result [3] that ascending sequences of uniformly r.e. degrees can have minimal upper bounds in the set R of r.e. degrees is another corollary of (2).

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 49 , Issue 2 , June 1984 , pp. 586 - 607
Copyright
Copyright © Association for Symbolic Logic 1984

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References

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