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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: hsimmons@manchester.ac.uk
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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

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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.CrossRefGoogle 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.CrossRefGoogle Scholar
[15] Smullyan, R. M., Theory of formal systems, Annals of Mathematics Studies, no. 47, Princeton University Press, 1961.Google Scholar