The typical Turing degree
  The Turing degree of a real measures the computational difficulty of producing its binary expansion. Since Turing degrees are tailsets, it follows from Kolmogorov's 0-1 law that for any property which may or may not be satisfied by any given Turing degree, the satisfying class will either be of Lebesgue measure 0 or 1, so long as it is measurable. So either the \emph{typical} degree satisfies the property, or else the typical degree satisfies its negation. Further, there is then some level of randomness sufficient to ensure typicality in this regard. A similar analysis can be made in terms of Baire category, where a standard form of genericity now plays the role that randomness plays in the context of measure.   We describe and prove a number of results in a programme of research which aims to establish the properties of the typical Turing degree, where typicality is gauged either in terms of Lebesgue measure or Baire category. 
George Barmpalias, Adam R. Day, Andrew E.M. Lewis
math.LO