Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-23T04:27:57.719Z Has data issue: false hasContentIssue false
  • English
  • Français

The Ackermann functions are not optimal, but by how much?

Published online by Cambridge University Press:  12 March 2014

H. Simmons
H. Simmons*
Affiliation:
School of Mathematics, The University, Manchester M13 9PL, England, E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

By taking a closer look at the construction of an Ackermann function we see that between any primitive recursive degree and its Ackermann modification there is a dense chain of primitive recursive degrees.

Keywords

Primitive recursive degreeAckermann jump
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 75 , Issue 1 , March 2010 , pp. 289 - 313
Copyright
Copyright © Association for Symbolic Logic 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1] Ackermann, W., Zum Hilbertschen Aufbau der reellen Zahlen, Mathematische Annalen, vol. 99 (1928), pp. 118–133.Google Scholar
[2] Cleave, J. P. and Rose, H. E., -arithmetic, Sets, models and recursion theory (Crossley, J. N., editor), North Holland, 1967, pp. 297–308.Google Scholar
[3] Grzegorczyk, A., Some classes of recursive functions, Rozprawy Matematyczne, vol. 4 (1953), pp. 1–45.Google Scholar
[4] Löb, M. E. and Wainer, S. S., Hierachies of number-theoretic functions, Part I, Archiv für Mathematische Logik und Grundlagenforschung, vol. 13 (1970), pp. 39–51.Google Scholar
[5] Löb, M. E. and Wainer, S. S., Hierachies of number-theoretic functions, Part II, Archiv für Mathematische Logik und Grundlagenforschung, vol. 13 (1970), pp. 97–113.Google Scholar
[6] McBeth, M. A., Combinatorial number theory, The Edwin Mellor Press, 1994.Google Scholar
[7] Odifreddi, P. G., Classical recursion theory, vol. I and II, Elsevier, 1999.Google Scholar
[8] Péter, R., Konstruktion nichtrekursiver Funktionen, Mathematische Annalen, vol. 111 (1935), pp. 42–60.Google Scholar
[9] Péter, R., Recursive functions, Academic Press, 1967.Google Scholar
[10] Robinson, R. M., Recursion and double recursion, Bulletin of the American Mathematical Society, vol. 54 (1948), pp. 987–993.Google Scholar
[11] Rose, H. E., Subrecursion, functions and hierarchies, Oxford University Press, 1984.Google Scholar
[12] Simmons, H., A density property of the primitive recursive degrees, Technical Report UMCS-93-1-1. Department Of Computer Science, Victoria University of Manchester.Google Scholar
[13] Simmons, H., Derivation and computation, Cambridge University Press, 2000.Google Scholar
[14] Smoryński, C., Logical number theory I, Springer, 1991.Google Scholar
[15] Smullyan, R. M., Theory of formal systems, Annals of Mathematics Studies, no. 47, Princeton University Press, 1961.Google Scholar