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Amenable equivalence relations and Turing degrees

Published online by Cambridge University Press:  12 March 2014

Alexander S. Kechris
Alexander S. Kechris*
Affiliation:
Department of Mathematics, California Institute of Technology, Pasadena, California 91125
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Extract

In [12] Slaman and Steel posed the following problem:

Assume ZF + DC + AD. Suppose we have a function assigning to each Turing degree d a linear order <d of d. Then must the rationals embed order preservingly in <d for a cone of d's?

They had already obtained a partial answer to this question by showing that there is no such d ↦ <d with <d of order type ζ = ω* + ω on a cone. Already the possibility that <d has order type ζ · ζ was left open.

We use here, ideas and methods associated with the concept of amenability (of groups, actions, equivalence relations, etc.) to prove some general results from which one can obtain a positive answer to the above problem.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 56 , Issue 1 , March 1991 , pp. 182 - 194
Copyright
Copyright © Association for Symbolic Logic 1991

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References

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