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Hypersimplicity and semicomputability in the weak truth table degrees

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Abstract

We study the classes of hypersimple and semicomputable sets as well as their intersection in the weak truth table degrees. We construct degrees that are not bounded by hypersimple degrees outside any non-trivial upper cone of Turing degrees and show that the hypersimple-free c.e. wtt degrees are downwards dense in the c.e. wtt degrees. We also show that there is no maximal (w.r.t. ≤wtt) hypersimple wtt degree. Moreover, we consider the sets that are both hypersimple and semicomputable, characterize them as the (bi-infinite) c.e. cuts of computable orderings of ℕ of order type ω+ω* and study their wtt degrees. We show that there are hypersimple degrees that are not bounded by any hypersimple semicomputable degree, investigate relationships with the join and look for maximal and minimal elements of related classes.

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  1. School of Mathematics, University of Leeds, Leeds, LS2 9JT, United Kingdom

    George Barmpalias

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  1. George Barmpalias
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Correspondence to George Barmpalias.

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I wish to thank the anonymous referee for making helpful remarks that have improved the presentation of this work.

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Barmpalias, G. Hypersimplicity and semicomputability in the weak truth table degrees. Arch. Math. Logic 44, 1045–1065 (2005). https://doi.org/10.1007/s00153-005-0288-9

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  • DOI: https://doi.org/10.1007/s00153-005-0288-9

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