Series: Penn State Logic Seminar

 Date: Tuesday, April 25, 2006

 Time: 2:30 - 3:45 PM

 Place: 106 McAllister Building

 Speaker: Esteban Gomez-Riviere, Penn State, Mathematics

 Title: Introduction to $K$-Trivial Reals, part 4


   In the previous talk we introduced the three properties of being
   strongly $K$-trivial, low-for-random, and basic-for-random and
   mentioned that they are equivalent.  We saw that being strongly
   $K$-trivial implies being low-for-random.  In this talk we complete
   the circle of equivalences.  We begin by proving Schnorr's Theorem
   which says that our notion of randomness is equivalent to
   Martin-Lof randomness.  We then prove the Kucera-Gacs theorem, from
   which it follows easily that being low for random implies being
   basic-for-random.  We then examine the proof that being
   basic-for-random implies being strongly $K$-trivial.  The
   usefulness of these equivalences comes from the fact that they are
   now known to be equivalent to $K$-triviality and each has different
   strengths in terms of proving properties of the class of
   $K$-trivial reals.