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A LIGHTFACE ANALYSIS OF THE DIFFERENTIABILITY RANK
Published online by Cambridge University Press: 17 April 2014
Abstract
We examine the computable part of the differentiability hierarchy defined by
Kechris and Woodin. In that hierarchy, the rank of a differentiable function is
an ordinal less than ${\omega _1}$ which measures how complex it is to verify differentiability
for 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 most
α is
${{\rm{\Pi }}_{2\alpha + 1}}$-complete. This result is expressed in the notation of Ash and
Knight.
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- Copyright © Association for Symbolic Logic 2014
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