Definable hereditary families in the projective hierarchy

We show that if ℱ is a hereditary family of subsets of ${\displaystyle {\omega }^{\omega }}$ satisfying certain definable conditions, then the ${\displaystyle {\Delta }_1^1}$ reals are precisely the reals α such that ${\displaystyle \{\beta :\alpha \in {\Delta }_1^1(\beta )\}\notin \in ℱ}$. This generalizes the results for measure and category. Appropriate generalization to the higher levels of the projective hierarchy is obtained under Projective Determinacy. Application of this result to the ${\displaystyle Q_{2n+1}}$-encodable reals is also shown.