Characterizing $E_{3}$ (the largest countable $\Pi ^{1}_{3}$ set)

Abstract:

Assume projective determinacy. For any real $\alpha$, let $\lambda _3^\alpha = \sup \{ \xi |\xi$ is the type of a prewellordering of the reals which is $\Delta _3^1$ in $\alpha \}$. Then, ${\mathcal {C}_3}$, the largest countable $\Pi _3^1$ set of reals, is equal to $\{ \alpha |\forall \beta (\lambda _3^\alpha \leqslant \lambda _3^\beta \Rightarrow \alpha$ is $\Delta _3^1$ in $\beta )\}$. This result, which is true for all odd levels and generalizes a previously known characterization of ${\mathcal {C}_1}$, answers a question of Kechris.