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A LIGHTFACE ANALYSIS OF THE DIFFERENTIABILITY RANK

Published online by Cambridge University Press:  17 April 2014

LINDA BROWN WESTRICK
LINDA BROWN WESTRICK*
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF CALIFORNIA-BERKELEY 970 EVANS HALL BERKELEY, CA 94720-3840, USA E-mail: westrick@math.berkeley.edu
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Abstract

We examine the computable part of the differentiability hierarchy defined byKechris and Woodin. In that hierarchy, the rank of a differentiable function isan ordinal less than ${\omega _1}$ which measures how complex it is to verify differentiabilityfor that function. We show that for each recursive ordinal $\alpha > 0$, the set of Turing indices of $C[0,1]$ functions that are differentiable with rank at mostα is ${{\rm{\Pi }}_{2\alpha + 1}}$-complete. This result is expressed in the notation of Ash andKnight.

Keywords

Differentiabilityhyperarithmetical hierarchyrank functioncomputable analysiseffective descriptive set theory

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Type
Articles
Information
The Journal of Symbolic Logic , Volume 79 , Issue 1 , March 2014 , pp. 240 - 265
Copyright
Copyright © Association for Symbolic Logic 2014 

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References

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