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Archive for Mathematical Logic

, Volume 41, Issue 6, pp 507–523 | Cite as

Post's problem for supertasks has both positive and negative solutions

  • Joel David Hamkins
  • Andrew Lewis

Abstract.

 The infinite time Turing machine analogue of Post's problem, the question whether there are semi-decidable supertask degrees between 0 and the supertask jump 0, has in a sense both positive and negative solutions. Namely, in the context of the reals there are no degrees between 0 and 0, but in the context of sets of reals, there are; indeed, there are incomparable semi-decidable supertask degrees. Both arguments employ a kind of transfinite-injury construction which generalizes canonically to oracles.

Keywords

Machine Analogue Turing Machine Infinite Time Negative Solution Infinite Time Turing Machine 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Joel David Hamkins
    • 1
  • Andrew Lewis
    • 2
  1. 1.Kobe University (Japan); and The College of Staten Island and the CUNY Graduate Center, 365 Fifth Avenue, New York, NY 10016, USA. e-mail: jhamkins@gc.cuny.eduUS
  2. 2.Department of Mathematical Sciences, Virginia Commonwealth University, P.O. Box 842014, Room 2078, Oliver Hall, 1015 West Main Street, Richmond, VA 23284-2014, USA. e-mail: amlewis@saturn.vcu.eduUS

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