An application of graphical enumeration to PA

Journal of Symbolic Logic 68 (1):5-16 (2003)
For α less than ε0 let $N\alpha$ be the number of occurrences of ω in the Cantor normal form of α. Further let $\mid n \mid$ denote the binary length of a natural number n, let $\mid n\mid_h$ denote the h-times iterated binary length of n and let inv(n) be the least h such that $\mid n\mid_h \leq 2$ . We show that for any natural number h first order Peano arithmetic, PA, does not prove the following sentence: For all K there exists an M which bounds the lengths n of all strictly descending sequences $\langle \alpha_0, ..., \alpha_n\rangle$ of ordinals less than ε0 which satisfy the condition that the Norm $N\alpha_i$ of the i-th term αi is bounded by $K + \mid i \mid \cdot \mid i\mid_h$ . As a supplement to this (refined Friedman style) independence result we further show that e.g., primitive recursive arithmetic, PRA, proves that for all K there is an M which bounds the length n of any strictly descending sequence $\langle \alpha_0,..., \alpha_n\rangle$ of ordinals less than ε0 which satisfies the condition that the Norm $N\alpha_i$ of the i-th term αi is bounded by $K + \mid i \mid\cdot inv(i)$ . The proofs are based on results from proof theory and techniques from asymptotic analysis of Polya-style enumerations. Using results from Otter and from Matou $\breve$ ek and Loebl we obtain similar characterizations for finite bad sequences of finite trees in terms of Otter's tree constant 2.9557652856...
Keywords No keywords specified (fix it)
Categories (categorize this paper)
DOI 10.2178/jsl/1045861503
 Save to my reading list
Follow the author(s)
My bibliography
Export citation
Find it on Scholar
Edit this record
Mark as duplicate
Revision history
Request removal from index
Download options
Our Archive

Upload a copy of this paper     Check publisher's policy     Papers currently archived: 28,126
Through your library
References found in this work BETA
Proof-Theoretic Investigations on Kruskal's Theorem.Michael Rathjen & Andreas Weiermann - 1993 - Annals of Pure and Applied Logic 60 (1):49-88.
Elementary Descent Recursion and Proof Theory.Harvey Friedman & Michael Sheard - 1995 - Annals of Pure and Applied Logic 71 (1):1-45.

Add more references

Citations of this work BETA
Unprovability Threshold for the Planar Graph Minor Theorem.Andrey Bovykin - 2010 - Annals of Pure and Applied Logic 162 (3):175-181.
Phase Transitions for Gödel Incompleteness.Andreas Weiermann - 2009 - Annals of Pure and Applied Logic 157 (2):281-296.
2004 Summer Meeting of the Association for Symbolic Logic.Wolfram Pohlers - 2005 - Bulletin of Symbolic Logic 11 (2):249-312.

View all 7 citations / Add more citations

Similar books and articles
Variations on a Theme by Weiermann.Toshiyasu Arai - 1998 - Journal of Symbolic Logic 63 (3):897-925.
On the T-Degrees of Partial Functions.Paolo Casalegno - 1985 - Journal of Symbolic Logic 50 (3):580-588.
Monotone Inductive Definitions in Explicit Mathematics.Michael Rathjen - 1996 - Journal of Symbolic Logic 61 (1):125-146.
Slim Models of Zermelo Set Theory.A. R. D. Mathias - 2001 - Journal of Symbolic Logic 66 (2):487-496.
Elementary Embedding Between Countable Boolean Algebras.Robert Bonnet & Matatyahu Rubin - 1991 - Journal of Symbolic Logic 56 (4):1212-1229.
Inequivalent Representations of Geometric Relation Algebras.Steven Givant - 2003 - Journal of Symbolic Logic 68 (1):267-310.
Undecidable Extensions of Skolem Arithmetic.Alexis Bés & Denis Richard - 1998 - Journal of Symbolic Logic 63 (2):379-401.

Monthly downloads

Added to index


Total downloads

26 ( #196,850 of 2,171,800 )

Recent downloads (6 months)

1 ( #326,702 of 2,171,800 )

How can I increase my downloads?

My notes
Sign in to use this feature

There  are no threads in this forum
Nothing in this forum yet.

Other forums