Series: Penn State Logic Seminar Date: Tuesday, April 11, 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 2 Abstract: In a previous talk we introduced $K$-trivial reals. In this talk we shall apply the Kraft-Chaitin theorem to prove Chaitin's 1975 result that all $K$-trivial reals are $\Delta^0_2$. Along the way we shall prove the following result of Zambella 1990: The number of strings of length $n$ of complexity at most $c$ plus the complexity of $n$ is $O(2^c)$. If time permits, we shall consider further properties of $K$-trivial reals, e.g., closure under join, and closure under Turing reducibility.