Abstract
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.
| Original language | English |
|---|---|
| Pages (from-to) | 249-253 |
| Number of pages | 5 |
| Journal | European Journal of Combinatorics |
| Volume | 9 |
| Issue number | 3 |
| DOIs | |
| Publication status | Published - 1988 |
| Externally published | Yes |
Fingerprint
ASJC Scopus subject areas
- Discrete Mathematics and Combinatorics
Cite this
Note on a Proof of the Extended Kirby—Paris Theorem on Labeled Finite Trees. / Okada, Mitsuhiro.
In: European Journal of Combinatorics, Vol. 9, No. 3, 1988, p. 249-253.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Note on a Proof of the Extended Kirby—Paris Theorem on Labeled Finite Trees
AU - Okada, Mitsuhiro
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.
UR - http://www.scopus.com/inward/record.url?scp=0013559538&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=0013559538&partnerID=8YFLogxK
U2 - 10.1016/S0195-6698(88)80016-7
DO - 10.1016/S0195-6698(88)80016-7
M3 - Article
VL - 9
SP - 249
EP - 253
JO - European Journal of Combinatorics
JF - European Journal of Combinatorics
SN - 0195-6698
IS - 3
ER -