## Journal of Symbolic Logic

### An application of graphical enumeration to PA*

Andreas Weiermann

#### 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 $N\ali$ of the $i$-th term $\ali$ 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 $N\ali$ of the $i$-th term $\ali$ 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... .

#### Article information

Source
J. Symbolic Logic, Volume 68, Issue 1 (2003), 5-16.

Dates
First available in Project Euclid: 21 February 2003

https://projecteuclid.org/euclid.jsl/1045861503

Digital Object Identifier
doi:10.2178/jsl/1045861503

Mathematical Reviews number (MathSciNet)
MR1959309

Zentralblatt MATH identifier
1041.03045

#### Citation

Weiermann, Andreas. An application of graphical enumeration to PA *. J. Symbolic Logic 68 (2003), no. 1, 5--16. doi:10.2178/jsl/1045861503. https://projecteuclid.org/euclid.jsl/1045861503

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