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http://dx.doi.org/10.14317/jami.2022.883

GOODSTEIN'S GENERALIZED THEOREM: FROM ROOTED TREE REPRESENTATIONS TO THE HYDRA GAME  

LEONARDIS, A. (Department of Mathematics and Computer Science, University of Calabria)
D'ATRI, G. (Department of Mathematics and Computer Science, University of Calabria)
ZANARDO, E. (Department of Digital Innovation, University of Nicosia)
Publication Information
Journal of applied mathematics & informatics / v.40, no.5_6, 2022 , pp. 883-896 More about this Journal
Abstract
A hereditary base-b representation, used in the celebrated Goodstein's theorem, can easily be converted into a labeled rooted tree. In this way it is possible to give a more elementary geometric proof of the aforementioned theorem and to establish a more general version, geometrically proved. This view is very useful for better understanding the underlying logical problems and the need to use transfinite induction in the proof. Similar problems will then be considered, such as the so-called "hydra game".
Keywords
Goodstein's theorem; rooted trees; unimaginable numbers; Knuth's up-arrow notation; number representation;
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