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An order-theoretic characterization of the Howard-Bachmann-hierarchy

(2017) ARCHIVE FOR MATHEMATICAL LOGIC. 56(1-2). p.79-118
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Abstract
In this article we provide an intrinsic characterization of the famous Howard-Bachmann ordinal in terms of a natural well-partial-ordering by showing that this ordinal can be realized as a maximal order type of a class of generalized trees with respect to a homeomorphic embeddability relation. We use our calculations to draw some conclusions about some corresponding subsystems of second order arithmetic. All these subsystems deal with versions of light-face Π11-comprehension.
Keywords
Kruskal’s theorem, Well-partial-orderings, Howard-Bachmann number, Ordinal notation systems, Natural well-orderings, Maximal order type, Collapsing function, Recursively defined trees, Tree-embedd abilities, Proof-theoretical ordinal, Impredicative theory, Independence results, Minimal bad sequence, WELL-PARTIAL-ORDERINGS, REVERSE MATHEMATICS

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Citation

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Chicago
Van der Meeren, Jeroen, Michael Rathjen, and Andreas Weiermann. 2017. “An Order-theoretic Characterization of the Howard-Bachmann-hierarchy.” Archive for Mathematical Logic 56 (1-2): 79–118.
APA
Van der Meeren, J., Rathjen, M., & Weiermann, A. (2017). An order-theoretic characterization of the Howard-Bachmann-hierarchy. ARCHIVE FOR MATHEMATICAL LOGIC, 56(1-2), 79–118.
Vancouver
1.
Van der Meeren J, Rathjen M, Weiermann A. An order-theoretic characterization of the Howard-Bachmann-hierarchy. ARCHIVE FOR MATHEMATICAL LOGIC. 2017;56(1-2):79–118.
MLA
Van der Meeren, Jeroen, Michael Rathjen, and Andreas Weiermann. “An Order-theoretic Characterization of the Howard-Bachmann-hierarchy.” ARCHIVE FOR MATHEMATICAL LOGIC 56.1-2 (2017): 79–118. Print.
@article{5836069,
  abstract     = {In this article we provide an intrinsic characterization of the famous Howard-Bachmann ordinal in terms of a natural well-partial-ordering by showing that this ordinal can be realized as a maximal order type of a class of generalized trees with respect to a homeomorphic embeddability relation. We use our calculations to draw some conclusions about some corresponding subsystems of second order arithmetic. All these subsystems deal with versions of light-face \ensuremath{\Pi}11-comprehension.},
  author       = {Van der Meeren, Jeroen and Rathjen, Michael and Weiermann, Andreas},
  issn         = {0933-5846},
  journal      = {ARCHIVE FOR MATHEMATICAL LOGIC},
  language     = {eng},
  number       = {1-2},
  pages        = {79--118},
  title        = {An order-theoretic characterization of the Howard-Bachmann-hierarchy},
  url          = {http://dx.doi.org/10.1007/s00153-016-0515-6},
  volume       = {56},
  year         = {2017},
}

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