Abstract
Concerning the set of rooted binary trees, one shows that Higman’s Lemma and Dershowitz’s recursive path ordering can be used for the decision of its maximal order type according to the homeomorphic embedding relation as well as of the order type according to its canonical linearization, well-known in proof theory as the Feferman-Schütte notation system without terms for addition. This will be done by showing that the ordinal ω n + 1 can be found as the (maximal) order type of a set in a cumulative hierarchy of sets of rooted binary trees.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
de Jongh, D.H.J., Parikh, R.: Well-partial orderings and hierarchies. Nederl. Akad. Wetensch. Proc. Ser. A 80=Indag. Math 39(3), 195–207 (1977)
Schmidt, D.: Well-Partial Orderings and Their Maximal order Types. Habilitationsschrift, Heidelberg (1979)
Higman, G.: Ordering by divisibility in abstract algebras. In: Proc. London Math. Soc 2(3), 326–336 (1952)
Hasegawa, R.: Well-ordering of algebras and Kruskal’s theorem. In: Sato, M., Hagiya, M., Jones, N.D. (eds.) Logic, Language and Computation. LNCS, vol. 792, pp. 133–172. Springer, Heidelberg (1994)
Feferman, S.: Systems of predicative analysis. J. Symbolic Logic 29, 1–30 (1964)
Feferman, S.: Systems of predicative analysis. II. Representations of ordinals. J. Symbolic Logic 33, 193–220 (1968)
Schütte, K.: Predicative well-orderings. In: Formal Systems and Recursive Functions. In: Proc. Eighth Logic Colloq, Oxford, 1963. North-Holland pp. 280–303 (1965)
Schütte, K.: Proof theory. In: Translated from the revised German edition by Crossley, J. N. Grundlehren der Mathematischen Wissenschaften, Band, Springer, Heidelberg (1977)
Weiermann, A.: Phase transition thresholds for some Friedman-style independence results (to appear)
Lee, G.: Slowly-well-orderedness of binary trees in Peano arithmetic and its fragments. (Preprint)
Rathjen, M., Weiermann, A.: Proof-theoretic investigations on Kruskal’s theorem. Ann. Pure Appl. Logic 60(1), 49–88 (1993)
Dershowitz, N.: Orderings for term-rewriting systems. Theoret. Comput. Sci. 17(3), 279–301 (1982)
Touzet, H.: A characterisation of multiply recursive functions with Higman’s lemma. Inform. and Comput, RTA ’99 (Trento) 178(2), 534–544 (2002)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Lee, G. (2007). Binary Trees and (Maximal) Order Types. In: Cooper, S.B., Löwe, B., Sorbi, A. (eds) Computation and Logic in the Real World. CiE 2007. Lecture Notes in Computer Science, vol 4497. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73001-9_48
Download citation
DOI: https://doi.org/10.1007/978-3-540-73001-9_48
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-73000-2
Online ISBN: 978-3-540-73001-9
eBook Packages: Computer ScienceComputer Science (R0)