## Journal of Symbolic Logic

### Variations on a Theme by Weiermann

Toshiyasu Arai

#### Abstract

Weiermann [18] introduces a new method to generate fast growing functions in order to get an elegant and perspicuous proof of a bounding theorem for provably total recursive functions in a formal theory, e.g., in PA. His fast growing function $\theta\alpha$n is described as follows. For each ordinal $\alpha$ and natural number n let T$^\alpha_n$ denote a finitely branching, primitive recursive tree of ordinals, i.e., an ordinal as a label is attached to each node in the tree so that the labelling is compatible with the tree ordering. Then the tree T$^\alpha_n$ is well founded and hence finite by Konig's lemma. Define $\theta\alpha$n=the depth of the tree T$^\alpha_n$=the length of the longest branch in T$^\alpha_n$. We introduce new fast and slow growing functions in this mode of definitions and show that each of these majorizes provably total recursive functions in PA.

#### Article information

Source
J. Symbolic Logic, Volume 63, Issue 3 (1998), 897-925.

Dates
First available in Project Euclid: 6 July 2007

https://projecteuclid.org/euclid.jsl/1183745573

Mathematical Reviews number (MathSciNet)
MR1649068

Zentralblatt MATH identifier
0919.03043

JSTOR