Archive for Mathematical Logic

, Volume 45, Issue 2, pp 179–190 | Cite as

The Medvedev lattice of computably closed sets

  • Sebastiaan A. Terwijn


Simpson introduced the lattice Open image in new window of Π01 classes under Medvedev reducibility. Questions regarding completeness in Open image in new window are related to questions about measure and randomness. We present a solution to a question of Simpson about Medvedev degrees of Π01 classes of positive measure that was independently solved by Simpson and Slaman. We then proceed to discuss connections to constructive logic. In particular we show that the dual of Open image in new window does not allow an implication operator (i.e. that Open image in new window is not a Heyting algebra). We also discuss properties of the class of PA-complete sets that are relevant in this context.

Keywords or phrases

Π01 classes Medvedev reducibility intuitionistic propositional logic 

Mathematics Subject Classification (2000)

03D30 03B55 03G10 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  1. 1.Institute of Discrete Mathematics and GeometryTechnical University of ViennaViennaAustria

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