Abstract

The Martin-Steel coarse inner model theory is employed in obtaining new results in descriptive set theory. $\underset{\sim}{\Pi}$ determinacy implies that for every thin $\Sigma^1_2$ equivalence relation there is a $\Delta^1_3$ real, $N$, over which every equivalence class is generic--and hence there is a good $\Delta^1_2$ wellordering of the equivalence classes. Analogous results are obtained for $\Pi^1_2$ and $\Delta^1_2$ quasilinear orderings and $\underset{\sim}{\Pi}^1_2$ determinacy is shown to imply that every $\Pi^1_2$ prewellorder has rank less than $\underset{\sim}{\delta}^1_2$.