Archive for Mathematical Logic

, Volume 56, Issue 5–6, pp 607–638 | Cite as

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

  • Michael Rathjen
  • Jeroen Van der Meeren
  • Andreas Weiermann


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 \(\varepsilon _0\). 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.


Well-partial-orderings Maximal order type Gap-embeddability relation Ordinal notation systems Collapsing function 

Mathematics Subject Classification

03F15 03E10 06A06 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

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

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