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Extensions of arithmetic for proving termination of computations

Published online by Cambridge University Press:  12 March 2014

Clement F. Kent  and
Bernard R. Hodgson
Clement F. Kent
Affiliation:
Department of Mathematical Sciences, Lakehead University, Thunder Bay, Ontario P7B 5E1, Canada
Bernard R. Hodgson
Affiliation:
Département de Mathématiques et de Statistique, Université Laval, Québec G1K 7P4, Canada
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Abstract

Kirby and Paris have exhibited combinatorial algorithms whose computations always terminate, but for which termination is not provable in elementary arithmetic. However, termination of these computations can be proved by adding an axiom first introduced by Goodstein in 1944. Our purpose is to investigate this axiom of Goodstein, and some of its variants, and to show that these are potentially adequate to prove termination of computations of a wide class of algorithms. We prove that many variations of Goodstein's axiom are equivalent, over elementary arithmetic, and contrast these results with those recently obtained for Kruskal's theorem.

Information

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 54 , Issue 3 , September 1989 , pp. 779 - 794
Copyright
Copyright © Association for Symbolic Logic 1989

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