TY - JOUR
T1 - Note on a Proof of the Extended Kirby—Paris Theorem on Labeled Finite Trees
AU - Okada, Mitsuhiro
N1 - Copyright:
Copyright 2017 Elsevier B.V., All rights reserved.
PY - 1988
Y1 - 1988
N2 - Buchholz [2] extended a certain game of unlabeled finite trees of Kirby—Paris [6] to the case of labeled finite trees whose nodes have labels from ω + 1 = {0, 1, 2, . . . , ω}, and proved that this game stops in finite time. He used an infinitary notion of ‘well-founded infinite trees’ to prove this property on the finite-tree game. In this note, we avoid the use of any infinitary notion and reduce the infinitary technique to a finitary technique, by utilizing Takeuti's system of ordinal diagrams [7]. Also we generalize Buchholz's game by introducing higher ordinal numbers as the labels of the trees, and show the termination property of this generalized game.
AB - Buchholz [2] extended a certain game of unlabeled finite trees of Kirby—Paris [6] to the case of labeled finite trees whose nodes have labels from ω + 1 = {0, 1, 2, . . . , ω}, and proved that this game stops in finite time. He used an infinitary notion of ‘well-founded infinite trees’ to prove this property on the finite-tree game. In this note, we avoid the use of any infinitary notion and reduce the infinitary technique to a finitary technique, by utilizing Takeuti's system of ordinal diagrams [7]. Also we generalize Buchholz's game by introducing higher ordinal numbers as the labels of the trees, and show the termination property of this generalized game.
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U2 - 10.1016/S0195-6698(88)80016-7
DO - 10.1016/S0195-6698(88)80016-7
M3 - Article
AN - SCOPUS:0013559538
VL - 9
SP - 249
EP - 253
JO - European Journal of Combinatorics
JF - European Journal of Combinatorics
SN - 0195-6698
IS - 3
ER -