Archive for Mathematical Logic

, Volume 56, Issue 1–2, pp 79–118 | Cite as

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

  • Jeroen Van der Meeren
  • Michael Rathjen
  • Andreas Weiermann


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 \(\varPi ^1_1\)-comprehension.


Well-partial-orderings Kruskal’s theorem Howard–Bachmann number Ordinal notation systems Natural well-orderings Maximal order type Collapsing function Recursively defined trees Tree-embeddabilities Proof-theoretical ordinal Impredicative theory Independence results Minimal bad sequence 

Mathematics Subject Classification

03B30 03E10 03E35 03F03 03F05 03F15 03F35 06A06 


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© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of MathematicsGhent UniversityGentBelgium
  2. 2.Department of Pure MathematicsUniversity of LeedsLeedsEngland, UK

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