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The ∀∃-theory of the effectively closed Medvedev degrees is decidable

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Abstract

We show that there is a computable procedure which, given an ∀∃-sentence \({\varphi}\) in the language of the partially ordered sets with a top element 1 and a bottom element 0, computes whether \({\varphi}\) is true in the Medvedev degrees of \({\Pi^0_1}\) classes in Cantor space, sometimes denoted by \({\mathcal{P}_s}\) .

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Authors and Affiliations

  1. Department of Mathematics, University of Notre Dame, 255 Hurley Hall, Notre Dame, IN, 46556-4618, USA

    Joshua A. Cole

  2. Mathematical Institute, Tohoku University, 6-3, Aramaki Aoba, Aoba-ku, Sendai, Miyagi, 980-8578, Japan

    Takayuki Kihara

Authors
  1. Joshua A. Cole
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  2. Takayuki Kihara
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Correspondence to Takayuki Kihara.

Additional information

Joshua Cole’s work is part of his PhD thesis at the University of Notre Dame under Peter Cholak. He was supported by a fellowship from the Arthur T. Schmitt Foundation and NSF-DMS-0652669 and NSF-EMSW21-RTG-0739007.

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Cole, J.A., Kihara, T. The ∀∃-theory of the effectively closed Medvedev degrees is decidable. Arch. Math. Logic 49, 1–16 (2010). https://doi.org/10.1007/s00153-009-0150-6

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  • DOI: https://doi.org/10.1007/s00153-009-0150-6

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