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Uniform upper bounds on ideals of turing degrees1

Published online by Cambridge University Press:  12 March 2014

Harold T. Hodes
Harold T. Hodes*
Affiliation:
Department of Philosophy, Harvard University, Cambridge, Massachusetts 02138 Department of Philosophy, Cornell University, Ithaca, New York 14850
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Given I, a reasonable countable set of Turing degrees, can we find some sort of canonical strict upper bound on I? If I = {a ∣ a ≤ b}, the upper bound on I which springs to mind is b′. But what if I is closed under jump? This question arises naturally out of the question which motivates a large part of hierarchy theory: Is there a canonical increasing function from a countable ordinal, preferably a large one, into D, the set of Turing degrees? If d is to be such a function, it is natural to require that d(α + 1) = d(α)′; but how should d(λ) depend on d ↾ λ, where λ is a limit ordinal?

For any I ⊆ D, let MI, = ⋃I. Towards making the above questions precise, we introduce ideals of Turing degrees.

Definition 1. I ⊆ D is an ideal iff I is closed under jump and join, and I is downward-closed, i.e., if a ≤ b & b ϵ I then a ϵ I.

The following definition reflects the hierarchy-theoretic motivation for this paper.

Definition 2. For I ⊆ D and A ⊆ ω, I is an A-hierarchy ideal iff for some countable ordinal α, MI = Lα[A]∩ ωω.

All hierarchy ideals are ideals, but not conversely.

Early in the game Spector knocked out the best sort of canonicity for upper bounds on ideals, proving that no set of degrees closed under jump has a least upper bound.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 43 , Issue 3 , September 1978 , pp. 601 - 612
Copyright
Copyright © Association for Symbolic Logic 1978

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Footnotes

1

Theorem 1, (1) ⇔ (2), and Theorem 2, (1*) ⇔ (2*), and possibly several other results reported here were obtained independently by Saul Kripke. Thanks to Andreas Blass, George Boolos, Carl Jockusch, Hilary Putnam and Steve Simpson for suggestions and encouragement.

References

BIBLIOGRAPHY

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