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Computers and the Mechanics of Communication Read chapter Read first chapter

2015 | OriginalPaper | Chapter

How to Compare Buchholz-Style Ordinal Notation Systems with Gordeev-Style Notation Systems

Authors : Jeroen Van der Meeren, Andreas Weiermann

Published in: Evolving Computability

Publisher: Springer International Publishing

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Abstract

By a syntactical construction we define an order-preserving mapping of Gordeev’s ordinal notation system \(PRJ(P)\) into Buchholz’s ordinal notation system \(OT(P)\) where \(P\) represents a limit ordinal. Since Gordeev already showed that \(OT(P)\) can be considered as a subsystem of \(PRJ(P)\), we obtain a direct proof of the equality of the order types of both systems. We expect that our result will contribute to the general program of determining the maximal order types of those well-quasi-orders which are provided by gap-embeddability relations considered by Friedman [10], Gordeev [5, 7] and Kriz [8].

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previous chapter Kalmár and Péter: Undecidability as a Consequence of Incompleteness
Literature
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Buchholz, W.: Normalfunktionen und konstruktive Systeme von Ordinalzahlen. In: \(\models \) ISILC Proof Theory Symposion (Proceedings of the International Summer Institute and Logic Colloquium, Kiel, 1974). Lecture Notes in Mathematics, vol. 500, pp. 4–25. Springer, Berlin (1975)
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Buchholz, W., Feferman, S., Pohlers, W., Sieg, W.: Iterated Inductive Definitions and Subsystems of Analysis: Recent Proof-theoretical Studies. Lecture Notes in Mathematics, vol. 897. Springer, Berlin (1981)
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Crossley, J.N., Kister, J.B.: Natural well-orderings. Arch. Math. Logik Grundlag. 26(1–2), 57–76 (1986/1987)
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Gordeev, L.: Generalizations of the one-dimensional version of the Kruskal-Friedman theorems. J. Symb. Log. 54(1), 100–121 (1989)MATHMathSciNetCrossRef
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Gordeev, L.: Systems of iterated projective ordinal notations and combinatorial statements about binary labeled trees. Arch. Math. Log. 29(1), 29–46 (1989)MATHMathSciNetCrossRef
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Kříž, I.: Well-quasiordering finite trees with gap-condition. Proof of Harvey Friedman’s conjecture. Ann. of Math. (2) 130(1), 215–226 (1989)MATHMathSciNetCrossRef
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Simpson, S.G.: Nonprovability of certain combinatorial properties of finite trees. In: Harrington, L.A., Morley, M.D., Ščedrov, A., Simpson, S.G. (eds.) Harvey Friedman’s Research on the Foundations of Mathematics. Studies in Logic and the Foundations of Mathematics, vol. 117, pp. 87–117. North-Holland, Amsterdam (1985)CrossRef
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Van der Meeren, J., Rathjen, M., Weiermann, A.: Well-partial-orderings and the big Veblen number. Arch. Math. Log. 54(1–2), 193–230 (2015)MATHCrossRef
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Weiermann, A.: A computation of the maximal order type of the term ordering on finite multisets. In: Ambos-Spies, K., Löwe, B., Merkle, W. (eds.) CiE 2009. LNCS, vol. 5635, pp. 488–498. Springer, Heidelberg (2009) CrossRef
Metadata
Title
How to Compare Buchholz-Style Ordinal Notation Systems with Gordeev-Style Notation Systems
Authors
Jeroen Van der Meeren
Andreas Weiermann
Copyright Year
2015
Book
Evolving Computability

Print ISBN: 978-3-319-20027-9

Electronic ISBN: 978-3-319-20028-6

Copyright Year: 2015

DOI
https://doi.org/10.1007/978-3-319-20028-6_36

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