March 2003 An application of graphical enumeration to PA*
Andreas Weiermann
J. Symbolic Logic 68(1): 5-16 (March 2003). DOI: 10.2178/jsl/1045861503

Abstract

For \al less than \eo let N\al be the number of occurrences of \om in the Cantor normal form of \al. Further let \lhn denote the binary length of a natural number n, let \lhhn denote the h-times iterated binary length of n and let \invn be the least h such that \lhhn2. 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 $\langle \al0,... ,\aln\rangleofordinalslessthan\eowhichsatisfytheconditionthattheNormN\alioftheithterm\aliisboundedbyK + \lh i\cdot \lhh{i}$.

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 $\langle \al0,... ,\aln\rangle ofordinalslessthan\eowhichsatisfiestheconditionthattheNormN\alioftheithterm\aliisboundedbyK + \lh i \cdot \inv i$. 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{\v{s}}ek and Loebl we obtain similar characterizations for finite bad sequences of finite trees in terms of Otter's tree constant 2.9557652856... .

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Andreas Weiermann. "An application of graphical enumeration to PA*." J. Symbolic Logic 68 (1) 5 - 16, March 2003. https://doi.org/10.2178/jsl/1045861503

Information

Published: March 2003
First available in Project Euclid: 21 February 2003

zbMATH: 1041.03045
MathSciNet: MR1959309
Digital Object Identifier: 10.2178/jsl/1045861503

Rights: Copyright © 2003 Association for Symbolic Logic

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Vol.68 • No. 1 • March 2003
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