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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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The complexity of recursion theoretic games
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by Martin Kummer PDF
Trans. Amer. Math. Soc. 358 (2006), 59-86 Request permission


We show that some natural games introduced by Lachlan in 1970 as a model of recursion theoretic constructions are undecidable, contrary to what was previously conjectured. Several consequences are pointed out; for instance, the set of all $\Pi _2$-sentences that are uniformly valid in the lattice of recursively enumerable sets is undecidable. Furthermore we show that these games are equivalent to natural subclasses of effectively presented Borel games.
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Additional Information
  • Martin Kummer
  • Affiliation: INIT GmbH, Kaeppelestrasse 6, D-76131 Karlsruhe, Germany
  • Email:
  • Received by editor(s): June 30, 2003
  • Published electronically: August 25, 2005
  • © Copyright 2005 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 358 (2006), 59-86
  • MSC (2000): Primary 03D25, 03D35, 03E15; Secondary 91A05, 91A46
  • DOI:
  • MathSciNet review: 2171223