Archive for Mathematical Logic

, Volume 41 , Issue 3 , pp 251 –257 | Cite as

A non-well-founded primitive recursive tree provably well-founded for co-r.e. sets

  • Arnold Beckmann


 We construct by diagonalization a non-well-founded primitive recursive tree, which is well-founded for co-r.e. sets, provable in Σ1 0. It follows that the supremum of order-types of primitive recursive well-orderings, whose well-foundedness on co-r.e. sets is provable in Σ1 0, equals the limit of all recursive ordinals ω1 ck . RID=""ID="" <E5>Mathematics Subject Classification (2000):&ensp;03B30</E5>, 03F15 RID=""ID="" Supported by the Deutschen Akademie der Naturforscher Leopoldina grant #BMBF-LPD 9801-7 with funds from the Bundesministerium f&uuml;r Bildung, Wissenschaft, Forschung und Technologie. RID=""ID="" I would like to thank A. SETZER for his hospitality during my stay in Uppsala in December 1998 &ndash; these investigations are inspired by a discussion with him; S. BUSS for his hospitality during my stay at UCSD and for valuable remarks on a previous version of this paper; and M. M&Ouml;LLERFELD for remarks on a previous title.


Subject Classification Previous Version Recursive Tree Recursive Ordinal Previous Title 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Arnold Beckmann
    • 1
  1. 1.Department of Mathematics, UCSD, 9500 Gilman Drive, La Jolla, CA 92093-0112, USA. e-mail: abeckman@math.ucsd.eduUS

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