Skip to main content



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.

Hide All
[1] Ash C. J. and Knight J., Computable structures and the hyperarithmetical hierarchy, Studies in Logic and the Foundations of Mathematics, vol. 144, North-Holland Publishing Co., Amsterdam, 2000. MR 1767842 (2001k:03090)
[2] Cenzer Douglas and Remmel Jeffrey B., Index sets for computable differential equations. Mathematical Logic Quarterly, vol. 50 (2004), no. 4–5, pp. 329344. MR 2090381 (2005h:03083)
[3] Greenberg Noam, Montalbán Antonio, and Slaman Theodore A., The Slaman-Wehner theorem in higher recursion theory. Proceedings of the American Mathematical Society, vol. 139 (2011), no. 5, pp. 18651869. MR 2763773 (2012c:03119)
[4] Grzegorczyk A., On the definitions of computable real continuous functions. Fundamenta Mathematicae, vol. 44 (1957), pp. 6171. MR 0089809 (19,723c)
[5] Ki Haseo, On the Denjoy rank, the Kechris-Woodin rank and the Zalcwasser rank. Transactions of the American Mathematical Society, vol. 349 (1997), no. 7, pp. 28452870. MR 1390042 (97i:04001)
[6] Kechris Alexander S. and Hugh Woodin W., Ranks of differentiable functions. Mathematika, vol. 33 (1986), no. 2, pp. 252278. MR 882498 (88d:03097)
[7] Lacombe Daniel, Extension de la notion de fonction récursive aux fonctions d’une ou plusieurs variables réelles. II, III. Comptes Rendus de l’Acad[start]é[end]mie des Sciences, vol. 241 (1955), pp. 1314, 151–153. MR 0072080 (17,225e)
[8] Lempp Steffen, Hyperarithmetical index sets in recursion theory. Transactions of the American Mathematical Society, vol. 303 (1987), no. 2, pp. 559583.
[9] Mazurkiewicz S., Über die Menge der differenzierbaren Funktionen.Fundamenta Mathematicae, vol. 27 (1936), pp. 244249.
[10] Myhill J., A recursive function, defined on a compact interval and having a continuous derivative that is not recursive. Michigan Mathematical Journal, vol. 18 (1971), pp. 9798. MR 0280373 (43 #6093)
[11] Pour-El Marian B. and Ian Richards J., Computability in analysis and physics , Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1989. MR 1005942 (90k:03062)
[12] Sacks Gerald E., Higher recursion theory , Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1990. MR 1080970 (92a:03062)
[13] Simpson Stephen G., Subsystems of second order arithmetic, second ed., Perspectives in Logic, Cambridge University Press, Cambridge, 2009. MR 2517689 (2010e:03073)
[14] Soare Robert I., Recursively enumerable sets and degrees , Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1987, A study of computable functions and computably generated sets. MR 882921 (88m:03003)
[15] Spector Clifford, Recursive well-orderings, this Journal, vol. 20 (1955), pp. 151–163. MR 0074347 (17,570b)
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
Please enter your name
Please enter a valid email address
Who would you like to send this to? *



Full text views

Total number of HTML views: 0
Total number of PDF views: 11 *
Loading metrics...

Abstract views

Total abstract views: 139 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 18th January 2018. This data will be updated every 24 hours.