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 Abstract: 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.