The strong Spector-Gandy Theorem for the higher analytical pointclasses

Abstract

Assuming projective determinacy, we extend Spector's strong version of the Spector-Gandy Theorem to all odd levels of the projective hierarchy: Theorem. For every space $X$ which is a finite product of the natural numbers $N$ and Baire space $N^N$ and for every n, if $P$ is a ${\Pi }_{2n+1}^1$ subset of $X$, then there is a ${\Pi }_{2n}^1$ set $Q$ such that $P(x)⟺(\exists !\alpha )Q(x,\alpha )⟺(\exists \alpha \in {\Delta }_{2n+1}^1(x))Q(x,\alpha )$.