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Conference on Computability in Europe

CiE 2009: Mathematical Theory and Computational Practice pp 11–20Cite as

Skolem + Tetration Is Well-Ordered

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Part of the Lecture Notes in Computer Science book series (LNTCS,volume 5635)

Abstract

The problem of whether a certain set of number-theoretic functions – defined via tetration (i.e. iterated exponentiation) – is well-ordered by the majorisation relation, was posed by Skolem in 1956. We prove here that indeed it is a computable well-order, and give a lower bound τ 0 on its ordinal.

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References

  1. Ehrenfeucht, A.: Polynomial functions with Exponentiation are well ordered. Algebra Universialis 3, 261–262 (1973)

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  2. Kruskal, J.: Well-quasi-orderings, the tree theorem, and Vazsonyi’s conjecture. Trans. Amer. Math. Soc. 95, 261–262 (1960)

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  3. Levitz, H.: An ordered set of arithmetic functions representing the least ε-number. Z. Math. Logik Grundlag. Math. 21, 115–120 (1975)

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  4. Levitz, H.: An initial segment of the set of polynomial functions with exponentiation. Algebra Universialis 7, 133–136 (1977)

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  5. Levitz, H.: An ordinal bound for the set of polynomial functions with exponentiation. Algebra Universialis 8, 233–243 (1978)

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  6. Richardson, D.: Solution to the identity problem for integral exponential functions. Z. Math. Logik Grundlag. Math. 15, 333–340 (1978)

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  7. Sierpiński, W.: Cardinal and Ordinal Numbers. PWN-Polish Scientific Publishers, Warszawa (1965)

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  8. Skolem, T.: An ordered set of arithmetic functions representing the least ε-number. Det Kongelige Norske Videnskabers selskabs Forhandlinger 29(12), 54–59 (1956)

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Authors and Affiliations

  1. Dept. of Mathematics, University of Oslo, P.B. 1053, Blindern, 0316, Oslo, Norway

    Mathias Barra & Philipp Gerhardy

Authors
  1. Mathias Barra
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  2. Philipp Gerhardy
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Editor information

Editors and Affiliations

  1. University of Heidelberg, Im Neuenheimer Feld 294, 69120, Heidelberg, Germany

    Klaus Ambos-Spies

  2. Universiteit van Amsterdam, Plantage Muidergracht 24, 1018, Amsterdam, TV, The Netherlands

    Benedikt Löwe

  3. Institut für Informatik, Ruprecht-Karls-Universität Heidelberg, Im Neuenheimer Feld 294, 69120, Heidelberg, Germany

    Wolfgang Merkle

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© 2009 Springer-Verlag Berlin Heidelberg

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Barra, M., Gerhardy, P. (2009). Skolem + Tetration Is Well-Ordered. In: Ambos-Spies, K., Löwe, B., Merkle, W. (eds) Mathematical Theory and Computational Practice. CiE 2009. Lecture Notes in Computer Science, vol 5635. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03073-4_2

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  • DOI: https://doi.org/10.1007/978-3-642-03073-4_2

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-03072-7

  • Online ISBN: 978-3-642-03073-4

  • eBook Packages: Computer ScienceComputer Science (R0)