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Variations on a Theme by Weiermann

Toshiyasu Arai
The Journal of Symbolic Logic
Vol. 63, No. 3 (Sep., 1998), pp. 897-925
DOI: 10.2307/2586719
Stable URL: http://www.jstor.org/stable/2586719
Page Count: 29
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Variations on a Theme by Weiermann
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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 θαn is described as follows. For each ordinal α and natural number n let Tαn 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 Tαn is well founded and hence finite by Konig's lemma. Define θαn=the depth of the tree Tαn=the length of the longest branch in Tαn. 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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