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# How to compare Buchholz-style ordinal notation systems with Gordeev-style notation systems

(2015) 9136. p.353-362
Author
Organization
Abstract
By a syntactical construction we define an order-preserving mapping of Gordeev’s ordinal notation system PRJ(P) into Buchholz ordinal nota- tion system OT(P) where P represents a limit ordinal. Since Gordeev already showed that OT(P ) can be considered as a subsystem of PRJ(P), we obtain a direct proof of the equality of the order types of both systems. We expect that our result will contribute to the general program of determining the maximal order types of those well-quasi orders which are provided by gap-embeddability relations considered by Friedman, Gordeev and Kriz.
Keywords
Gordeev’s notation system, Order type, Buchholz’s notation system, Ordinal notation systems, Embeddings, Well-partial-orderings

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## Citation

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Chicago
Van der Meeren, Jeroen, and Andreas Weiermann. 2015. “How to Compare Buchholz-style Ordinal Notation Systems with Gordeev-style Notation Systems.” In Lecture Notes in Computer Science, ed. A Beckmann, V Miltrana, and M Soskova, 9136:353–362. Cham, Switzerland: Springer.
APA
Van der Meeren, J., & Weiermann, A. (2015). How to compare Buchholz-style ordinal notation systems with Gordeev-style notation systems. In A Beckmann, V. Miltrana, & M. Soskova (Eds.), Lecture Notes in Computer Science (Vol. 9136, pp. 353–362). Presented at the 11th Conference on Computability in Europe (CiE 2015), Cham, Switzerland: Springer.
Vancouver
1.
Van der Meeren J, Weiermann A. How to compare Buchholz-style ordinal notation systems with Gordeev-style notation systems. In: Beckmann A, Miltrana V, Soskova M, editors. Lecture Notes in Computer Science. Cham, Switzerland: Springer; 2015. p. 353–62.
MLA
Van der Meeren, Jeroen, and Andreas Weiermann. “How to Compare Buchholz-style Ordinal Notation Systems with Gordeev-style Notation Systems.” Lecture Notes in Computer Science. Ed. A Beckmann, V Miltrana, & M Soskova. Vol. 9136. Cham, Switzerland: Springer, 2015. 353–362. Print.
```@inproceedings{5836079,
abstract     = {By a syntactical construction we define an order-preserving mapping of Gordeev{\textquoteright}s ordinal notation system PRJ(P) into Buchholz ordinal nota- tion system OT(P) where P represents a limit ordinal. Since Gordeev already showed that OT(P ) can be considered as a subsystem of PRJ(P), we obtain a direct proof of the equality of the order types of both systems. We expect that our result will contribute to the general program of determining the maximal order types of those well-quasi orders which are provided by gap-embeddability relations considered by Friedman, Gordeev and Kriz.},
author       = {Van der Meeren, Jeroen and Weiermann, Andreas},
booktitle    = {Lecture Notes in Computer Science},
editor       = {Beckmann, A and Miltrana, V and Soskova, M},
isbn         = {9783319200286},
issn         = {0302-9743},
language     = {eng},
location     = {Bucharest, Romania},
pages        = {353--362},
publisher    = {Springer},
title        = {How to compare Buchholz-style ordinal notation systems with Gordeev-style notation systems},
url          = {http://dx.doi.org/10.1007/978-3-319-20028-6\_36},
volume       = {9136},
year         = {2015},
}

```
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