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

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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.

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Authors and Affiliations

  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.edu, , , , , , US

    Joel David Hamkins

  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.edu, , , , , , US

    Andrew Lewis

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  1. Joel David Hamkins
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  2. Andrew Lewis
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Received: 31 August 1998 / Published online: 12 July 2002

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Hamkins, J., Lewis, A. Post's problem for supertasks has both positive and negative solutions. Arch. Math. Logic 41, 507–523 (2002). https://doi.org/10.1007/s001530100112

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  • DOI: https://doi.org/10.1007/s001530100112

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