Series: Penn State Logic Seminar Date: Tuesday, November 15, 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 5 Abstract: In this talk I shall discuss the arithmetical hierarchy over the Baire space, omega^omega, noting in particular its close relationship with the hierarchy of open sets, closed sets, G_delta sets, F_sigma sets, ..., from descriptive set theory. In addition, I shall point out that the arithmetical hierarchy over the Baire space behaves somewhat differently from the arithmetical hierarchy over the integers. For instance, Post's Theorem fails in this context. After that, I shall begin my discussion of basis theorems and mass problems. Loosely speaking, the Basis Problem reads as follows. Given a "tame" subset P of the Baire space which is nonempty, to find a "tame" member of P. I shall present various solutions to the Basis Problem in the case when P is a recursively bounded Pi^0_1 subset of the Baire space.