Abstract
For less than let be the number of occurrences of in the Cantor normal form of . Further let denote the binary length of a natural number , let denote the -times iterated binary length of and let be the least such that . We show that for any natural number first order Peano arithmetic, , does not prove the following sentence: For all there exists an which bounds the lengths of all strictly descending sequences $\langle \al0,... ,\aln\rangle\eoN\alii\aliK + \lh i\cdot \lhh{i}$.
As a supplement to this (refined Friedman style) independence result we further show that e.g., primitive recursive arithmetic, , proves that for all there is an which bounds the length of any strictly descending sequence $\langle \al0,... ,\aln\rangle \eoN\alii\aliK + \lh i \cdot \inv i$. 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... .
Citation
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
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