Series: Penn State Logic Seminar Date: Tuesday, December 6, 2005 Time: 2:30 - 3:45 PM Place: 106 McAllister Building Speaker: Stephen G. Simpson, Penn State, Mathematics Title: An Introduction to Degrees of Unsolvability, part 7 Abstract: In this talk we introduce the idea of mass problems, which goes back to Kolmogorov. Informally, a mass problem is a problem which may have more than one solution. Formally, a mass problem is any subset P of the Baire space. The "solutions" of P are the members of P, and to "solving" P means finding a member of P. (For example, consider the following mass problem: to find a complete extension of Peano Arithmetic. This problem is unsolvable, in the sense that there is no complete extension of Peano Arithmetic which is computable. We may identify this problem with the set of complete extensions of Peano Arithmetic.) If P and Q are mass problems, we say that P is weakly reducible to Q if for every g in Q there exists f in P such that f is Turing reducible to Q. We say that P and Q are weakly equivalent if each is weakly reducible to the other. The weak degree of P is the equivalence class of P under weak equivalence. The weak degrees have an obvious partial ordering induced by weak reducibility. The partial ordering of weak degrees includes the partial ordering of Turing degrees, because we may identify the Turing degree of f with the weak degree of the singleton set {f}. We note some applications of Basis Theorems to the study of weak degrees and mass problems.