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Variations on a theme by Weiermann

Published online by Cambridge University Press:  12 March 2014

Toshiyasu Arai
Toshiyasu Arai*
Affiliation:
Faculty of Integrated Arts and Sciences, Hiroshima University, Higashi-Hiroshima, 739-8521, Japan E-mail: arai@mis.hiroshima-u.ac.jp
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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 θαη is described as follows. For each ordinal a and natural number n let 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 is well founded and hence finite by König's lemma. Define θαηn =the depth of the tree =the length of the longest branch in .

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.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 63 , Issue 3 , September 1998 , pp. 897 - 925
Copyright
Copyright © Association for Symbolic Logic 1998

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