Journal of Symbolic Logic

Variations on a Theme by Weiermann

Toshiyasu Arai

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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.

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

First available in Project Euclid: 6 July 2007

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Arai, Toshiyasu. Variations on a Theme by Weiermann. J. Symbolic Logic 63 (1998), no. 3, 897--925.

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