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Bounds for the closure ordinals of essentially monotonic increasing functions

Published online by Cambridge University Press:  12 March 2014

Andreas Weiermann
Andreas Weiermann*
Affiliation:
Institut für Mathematische Logik und Grundlagenforschung, Westfälische, Wilhelms-Universität Münster, 48149 Munster, Germany, E-mail: weiermanemath.uni-muenster.de
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Abstract

Let Ω ≔ ℵ1. For any α< εΩ + 1 ≔ min {ξ > Ω: ξ = ωξ} let EΩ(α) be the finite set of ε-numbers below Ω which are needed for the unique representation of α in Cantor-normal form using 0, Ω, +, and ω. Let α* ≔ max(EΩ(α) ∪ {0}). A function f: εΩ + 1 → Ω is called essentially increasing, if for any α < εΩ + 1; f(α) ≥ α*: f is called essentially monotonic, if for any α, β<εΩ + 1;

Let be the least set of ordinals which contains 0 as an element and which satisfies the following two conditions:

(a)

(b)

Let be the Howard-Bachmann ordinal, which is, for example, defined in [3]. The following theorem is shown: If f: εΩ + 1 → Ω is essentially monotonic and essentially increasing, then the order type of is less than or equal to .

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 58 , Issue 2 , June 1993 , pp. 664 - 671
Copyright
Copyright © Association for Symbolic Logic 1993

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References

REFERENCES

[1]Kister, J. and Crossley, J. N., Natural well-orderings, Archiv für Mathematische Logik und Grundlagenforschung, vol. 26 (1987), pp. 57–76.Google Scholar
[2]Feferman, S., Hereditarily replete functionals over the ordinals, Intuitionism and proof theory, proceedings of the summer conference at Buffalo, New York, 1968 (Kino, A., Myhill, J., Vesley, R. E., editors), North-Holland Publishing Company, Amsterdam, London, 1970.Google Scholar
[3]Pohlers, W., Proof theory, an introduction, Lecture Notes in Mathematics, vol. 1407, Springer-Verlag, Berlin and New York, 1989.Google Scholar
[4]Schmidt, D., Bounds for the closure ordinals of replete monotonic increasing functions, this Journal, vol. 40 (1975), pp. 305–316.Google Scholar
[5]Schmidt, D., Well-Partial orderings and their maximal order types, Habilitationsschrift, Ruprecht-Karl-Universität, Heidelberg, 1979.Google Scholar