A comparison of well-known ordinal notation systems for epsilon0

Abstract We consider five ordinal notation systems of e 0 which are all well-known and of interest in proof-theoretic analysis of Peano arithmetic: Cantor’s system, systems based on binary trees and on countable tree-ordinals, and the systems due to Schutte and Simpson, and to Beklemishev. The main point of this paper is to demonstrate that the systems except the system based on binary trees are equivalent as structured systems, in spite of the fact that they have their origins in different views and trials in proof theory. This is true while Weiermann’s results based on Friedman-style miniaturization indicate that the system based on binary trees is of different character than the others.

[1]  Stephen G. Simpson,et al.  Nonprovability of Certain Combinatorial Properties of Finite Trees , 1985 .

[2]  Toshiyasu Arai.  On the Slowly Well Orderedness of epsilon~0 , 2002 .

[3]  Graham Higman,et al.  Ordering by Divisibility in Abstract Algebras , 1952 .

[4]  G. Boolos.  The logic of provability , 1993 .

[5]  Andreas Weiermann,et al.  A Uniform Approach to Fundamental Sequences and Hierarchies , 1994, Math. Log. Q..

[6]  Stanley S. Wainer,et al.  Chapter III - Hierarchies of Provably Recursive Functions , 1998 .

[7]  G. Gentzen.  Die Widerspruchsfreiheit der reinen Zahlentheorie , 1936 .

[8]  Lev D. Beklemishev.  The Worm principle , 2003 .

[9]  Toshiyasu Arai.  On the Slowly Well Orderedness of epsilon0 , 2002, Math. Log. Q..

[10]  Michael Rathjen,et al.  Proof-Theoretic Investigations on Kruskal's Theorem , 1993, Ann. Pure Appl. Log..

[11]  Lorenzo Carlucci.  Worms, gaps, and hydras , 2005, Math. Log. Q..

[12]  J. Girard.  Proof Theory and Logical Complexity , 1989 .

[13]  J. Kruskal.  Well-quasi-ordering, the Tree Theorem, and Vazsonyi’s conjecture , 1960 .

[14]  Solomon Feferman,et al.  Systems of predicative analysis, II: Representations of ordinals , 1968, Journal of Symbolic Logic.

[15]  S. Shelah,et al.  Annals of Pure and Applied Logic , 1991 .

[16]  Wilfried Buchholz,et al.  A new system of proof-theoretic ordinal functions , 1986, Ann. Pure Appl. Log..

[17]  R. Otter.  The Number of Trees , 1948 .

[18]  J. Paris,et al.  Accessible Independence Results for Peano Arithmetic , 1982 .

[19]  Rick L. Smith.  The Consistency Strengths of Some Finite Forms of the Higman and Kruskal Theorems , 1985 .

[20]  Solomon Feferman,et al.  Systems of predicative analysis , 1964, Journal of Symbolic Logic.

[21]  K. Schütte,et al.  Predicative Well-Orderings , 1965 .

[22]  Lev D. Beklemishev,et al.  Provability algebras and proof-theoretic ordinals, I , 2001, Ann. Pure Appl. Log..

[23]  Ryu Hasegawa,et al.  Well-Ordering of Algebras and Kruskal's Theorem , 1994, Logic, Language and Computation.

[24]  Dick H. J. Jongh,et al.  Well-partial orderings and hierarchies , 1977 .

[25]  Andreas Weiermann.  Analytic combinatorics, proof-theoretic ordinals, and phase transitions for independence results , 2005, Ann. Pure Appl. Log..

[26]  Andreas Weiermann,et al.  An application of graphical enumeration to PA* , 2003, Journal of Symbolic Logic.

[27]  Andreas Weiermann,et al.  Phasenübergänge in Logik und Kombinatorik , 2005 .

[28]  Martin Loebl,et al.  On undecidability of the weakened Kruskal Theorem , 1987 .

[29]  Andreas Weiermann.  Analytic combinatorics for a certain well-ordered class of iterated exponential terms , 2005 .

[30]  Akshay Chandra Dey,et al.  Worms , 1919, The Indian medical gazette.

[31]  Stephen G. Simpson,et al.  Ein in der reinen Zahlentheorie unbeweisbarer Satz über endliche Folgen von natürlichen Zahlen , 1985, Arch. Math. Log..