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Ordinal notation systems corresponding to Friedman's linearized well-partial-orders with gap-condition

(2017) ARCHIVE FOR MATHEMATICAL LOGIC. 56(5-6). p.607-638
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Abstract
In this article we investigate whether the following conjecture is true or not: does the addition-free theta functions form a canonical notation system for the linear versions of Friedman's well-partial-orders with the so-called gap-condition over a finite set of n labels. Rather surprisingly, we can show this is the case for two labels, but not for more than two labels. To this end, we determine the order type of the notation systems for addition-free theta functions in terms of ordinals less than . We further show that the maximal order type of the Friedman ordering can be obtained by a certain ordinal notation system which is based on specific binary theta functions.
Keywords
Well-partial-orderings, Maximal order type, Gap-embeddability relation, Ordinal notation systems, Collapsing function

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Chicago
Rathjen, Michael, Jeroen Van der Meeren, and Andreas Weiermann. 2017. “Ordinal Notation Systems Corresponding to Friedman’s Linearized Well-partial-orders with Gap-condition.” Archive for Mathematical Logic 56 (5-6): 607–638.
APA
Rathjen, M., Van der Meeren, J., & Weiermann, A. (2017). Ordinal notation systems corresponding to Friedman’s linearized well-partial-orders with gap-condition. ARCHIVE FOR MATHEMATICAL LOGIC, 56(5-6), 607–638.
Vancouver
1.
Rathjen M, Van der Meeren J, Weiermann A. Ordinal notation systems corresponding to Friedman’s linearized well-partial-orders with gap-condition. ARCHIVE FOR MATHEMATICAL LOGIC. 2017;56(5-6):607–38.
MLA
Rathjen, Michael, Jeroen Van der Meeren, and Andreas Weiermann. “Ordinal Notation Systems Corresponding to Friedman’s Linearized Well-partial-orders with Gap-condition.” ARCHIVE FOR MATHEMATICAL LOGIC 56.5-6 (2017): 607–638. Print.
@article{8524897,
  abstract     = {In this article we investigate whether the following conjecture is true or not: does the addition-free theta functions form a canonical notation system for the linear versions of Friedman's well-partial-orders with the so-called gap-condition over a finite set of n labels. Rather surprisingly, we can show this is the case for two labels, but not for more than two labels. To this end, we determine the order type of the notation systems for addition-free theta functions in terms of ordinals less than . We further show that the maximal order type of the Friedman ordering can be obtained by a certain ordinal notation system which is based on specific binary theta functions.},
  author       = {Rathjen, Michael and Van der Meeren, Jeroen and Weiermann, Andreas},
  issn         = {0933-5846},
  journal      = {ARCHIVE FOR MATHEMATICAL LOGIC},
  keyword      = {Well-partial-orderings,Maximal order type,Gap-embeddability relation,Ordinal notation systems,Collapsing function},
  language     = {eng},
  number       = {5-6},
  pages        = {607--638},
  title        = {Ordinal notation systems corresponding to Friedman's linearized well-partial-orders with gap-condition},
  url          = {http://dx.doi.org/10.1007/s00153-017-0559-2},
  volume       = {56},
  year         = {2017},
}

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