Journal of Symbolic Logic

An application of graphical enumeration to PA*

Andreas Weiermann

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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

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

First available in Project Euclid: 21 February 2003

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


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

Export citation


  • T. Arai On the slowly well orderedness of $\varepsilon_0$, Mathematical Logic Quarterly, vol. 48 (2002), pp. 125--130.
  • S. N. Burris Number theoretic density and logical limit laws, Mathematical Surveys and Monographs, vol. 86, American Mathematical Society.
  • H. Friedman and M. Sheard Elementary descent recursion and proof theory, Annals of Pure and Applied Logic, vol. 71 (1995), pp. 1--45.
  • G. H. Hardy and S. Ramanujan Asymptotic formulae for the distribution of integers of various types, Proceedings of the London Mathematical Society, vol. 16 (1917), pp. 112--132.
  • J. B. Kruskal Well-quasi-orderings, the tree theorem, and Vázsonyi's conjecture, Transactions of the American Mathematical Society, vol. 95 (1960), pp. 210--225.
  • M. Loebl and J. Matoušek On undecidability of the weakened Kruskal theorem, Logic and combinatorics (Arcata, California, 1985), American Mathematical Society, Providence, RI,1987, pp. 275--280.
  • J. Matoušek and J. Nešetřil Invitation to Discrete Mathematics, The Clarendon Press, Oxford, New York,1998.
  • D. J. Newman Analytic Number Theory, Springer, New York,1998.
  • R. Otter The number of trees, Annals of Mathematics, vol. 49 (1948), pp. 583--599.
  • M. Rathjen and A. Weiermann Proof-theoretic investigations on Kruskal's theorem, Annals of Pure and Applied Logic, vol. 60 (1993), pp. 49--88.
  • J. Riordan The enumeration of trees by height and diameter, IBM Journal of Research and Development, vol. 4 (1960), pp. 473--478.
  • K. Schütte Proof Theory, Springer, Berlin,1977.
  • S. G. Simpson Non-provability of certain combinatorial properties of finite trees, Harvey Friedman's research on the foundations of mathematics, North-Holland, Amsterdam,1985, pp. 87--117.
  • R. Smith The consistency strength of some finite forms of the Higman and Kruskal theorems, Harvey Friedman's research on the foundations of mathematics, North-Holland, Amsterdam,1985, pp. 119--136.
  • A. Weiermann What makes a (point-wise) subrecursive hierarchy slow growing?, Sets and proofs (Leeds, 1997), Cambridge University Press, Cambridge,1999, pp. 403--423.
  • M. Yamashita Asymptotic estimation of the number of trees, Transactions of the Institute of Electronics and Communication Engineering of Japan, vol. 62-A (1979), pp. 128--135, in Japanese.