An application of graphical enumeration to PA*

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 $\lh n$ denote the binary length of a natural number $n$, let $\lhh n$ denote the $h$-times iterated binary length of $n$ and let $\inv n$ be the least $h$ such that $\lhh n \leq2$. 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 $\langle \al₀,... ,\aln\rangle$ of ordinals less than $\eo$ which satisfy the condition that the Norm $N\al_i$ of the $i$-th term $\al_i$ is bounded by $K + \lh i\cdot \lhh{i}$.

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 $ of ordinals less than $\eo$ which satisfies the condition that the Norm $N\al_i$ of the $i$-th term $\al_i$ is bounded by $K + \lh i \cdot \inv i$. 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... .