Some applications of coarse inner model theory

The Martin-Steel coarse inner model theory is employed in obtaining new results in descriptive set theory. determinacy implies that for every thin equivalence relation there is a real, N , over which every equivalence class is generic—and hence there is a good ( N # ) wellordering of the equivalence classes. Analogous results are obtained for and quasilinear orderings and determinacy is shown to imply that every prewellorder has rank less than .